59

òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

  • Upload
    others

  • View
    0

  • Download
    0

Embed Size (px)

Citation preview

Page 1: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

� � ��� ����

� ��� ���� � ����������� ��� � ��� �� � ��������� !�"���$#�� #&% � ��� '���( ) �* +�,

-/.1020435076984043;:70=<?>[email protected]>[email protected][Z]\=^`_ba�cedgfbhbi5j

k�lYmonbpHqrmstavuxwyZPz{\`Z}|A~�_b�y^���w1�9z�uxZ]\���Z��yZg�9��u���~�u`w����g�A^r_9����Zg��\`ay~�ay�2�g���y^x�9��\`Z]����ux~�_ba��^`w{~�zy^P�G\`_9X�\x�g�"�1�Lu��{�Fsta�u`wy~�^P��wy�9z{uxZ}\S��Z�Z���zy��_b\=Z7uxwyZ7�����A^�_9����Zg�9\=ay~�ay�^`�{��w7\`Z]����ux~�_bay^=wy~�zy^��G\`_9X�� �]_9X2¡y~�ay��ux~�_ba�_��1�1��ux�o�9ay�2¢?�1����~�ux��ux~�|bZ£�g���y^x�9���9^O�^`�{X¤z{u`~�_ba{^¥u`w1��u��9\`Z �yZ]Z}X¤Z}��zy�¦���y^`~�¡y��Z�~�a����b~�|bZ}a��y_bX¤�9~�a��Y§¨_b\=Z©¡y\=_5�9�y���bªuxw{~�^¥��w1�9z{u`Z]\��9~�X¤^£ux_�wyZ]��z�\`Z}^`Z]�9\`��wyZ}\`^��}_bX¤XH�yay~��]��uxZr¢?�1�9��~�u��Lux~�|9Zr�9^`^=�yX¤zA�ux~�_bay^«�9¡¬_9�{u«�]�9�y^=Z4�tZ4­�Z}�}u�\=Z]�¦�Lux~�_9ay^`wy~�zy^]ªPZ}���y�}~��1�LuxZ�u`wyZM\`�9X[~�®1�]��ux~�_bay^¯_9�^`�{��wM�9^=^`�yX[z{u`~�_bay^}ª°�9ay���yZ}\`~�|bZ7�g���y^x�9��~�a{�GZ}\`Z]a{�]Z]^o�G\=_bX&���]_bX2¡{~�a1��u`~�_ba¨_9��9^=^`�yX[z{ux~�_bay^}ª1Z4��z¬Z}\`~�X¤Z}a?u`^]ª±�9a{�¨�1�Lu��{�¥²o�y\�X¤�;³=_b\�ux�9^`´���~����¬¡¬ZPux_F�yZ}�]~��yZ��wyZ4uxwyZ}\ru`wyZS�9^`^=�yX¤z�ux~�_9ay^��b~�|9Z]aµ�9\=ZS^=�{¶[�]~�Z]a5u��G_b\��9^`^=Z]^=^`~�a{�[u`wyZ�^=ux\=Z]ay��uxw_9�$�g�9�{^x�9�°Z4­�Z}�}ux^��G\`_9X�ay_bayZ���z¬Z}\`~�X[Z]a5ux�9�/�1��ux�{�

· �9�y^`�9�2Z}­¬Z]�4ux^¸zUZ]\=X¤~�u«�{^¸ux_+zy\=Z]�y~��}u«wy_��¹^e��^euxZ}X¤^¸��_b�y���º\`Z}^`zU_bay�ux_[w5�Az¬_9u`wyZ}u`~��]�9�°~�a5uxZ}\=|9Z]a5ux~�_bay^O»+�G_b\�Z4�{�9X[zy��Z�ª1z¬_9��~��4���yZ}�]~�^`~�_9ay^�_b\��9�}u`~�_ba{^zUZ]\=�G_9\`X[Z]��~�a�Z4|bZ}\=�A�1�g�¼�9�}u`~�|A~�ut�9��½©^2��Zµwy��|9Z¯^`Z]Z}a¾~�a · w1�9z�uxZ]\vd�cÀ¿�Z]���ux~�_ba¾db�ÂÁ5j4ª�^`�y��w�zy\`Z}�y~��4ux~�_bay^S�9\`ZFuxw{Z[wy�9����X��9\=´¨_����g���y^x�9�£X[_��{Z]��~�ay�yª°^=~�ay�}Z

d�ibÁ

Page 2: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d�i9Ã1ÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Êuxw{Z}�v��\`Zoay_�u��y~�^`�]Z}\`ay~�¡y��Z �G\`_bXÚzy\=_b¡1�9¡y~���~�^=ux~���~�a{�G_b\=X��Lux~�_9a��9��_bayZ�Û�uxw{Z}�¤\`Z]^eu_baA»Ü�9ay�/ª5~�a2�D�9�}u]ª5�yZ}®yayZ�»Ý�g�9�y^`�9�±\`Z}�¦��u`~�_bay^=wy~�z{^]��Þ�wy~�^Y��w1��z{uxZ}\¥�{^`Z]^£�]�9�y^x����y~��9�b\x��X¤^£ux_7�b~�|bZ©�G_9\`X¤�9�±^`Z]X¤�9a5ux~��]^�u`_HuxwyZPa{_9ux~�_ba�_9�£IQK1Bt04ß�à904K1BÀIJ6LK1ª{�9ay�v~�uzy\=_;|A~��yZ]^�Z���zy��~��]~�u£�G_b\=X2�y���9^¥�G_9\�zU_b^=u`~�a5uxZ}\=|9Z]a5ux~�_ba�zy\=_b¡1�9¡y~���~�ux~�Z}^Y~�a�u`Z]\=X¤^¥_9�zy\=Z]~�a5u`Z]\=|9Z]a5ux~�_ba«z{\`_b¡1��¡y~���~�ux~�Z]^}�ÆÞ�wyZS~�X[zy��~��g��u`~�_ba[~�^�uxwy��u�uxwyZ�Z}­¬Z]�4ux^r_9�ÆZ4|5�Z]\e�2~�a?u`Z]\e|bZ]a5u`~�_ba7�]�9a7¡¬Z�Z]^=u`~�X¤��uxZ}�2�G\`_bX�ay_9ayZ4��zUZ]\`~�X¤Z}a5u��9���1��u���ªbzy\`_�|A~��{Z]�uxw{Z��1��ux�©~�^�^`�{zyzy��Z}X¤Z}a5uxZ]�2��~�uxwH� �g���y^x�9���y~¦���b\x�9Xáuxw1��u�~�^�¡¬_�uxw7�9�4�A�]��~�����ay��]_9a?ux�9~�a{^ra{_[����uxZ}a5u�|9��\`~¦��¡y��Z}^]�

sâ��^`_bX[Z¯|9��\`~¦��¡y��Z}^��9\=Z¨a{_9u�X[Zg�9^=�y\`Z}�ãuxw{Z]aãu`wyZ«¢?�yZ}^=ux~�_ba+_9�o~��yZ}a?u`~�®{��9¡y~���~�ut�¾�9\`~�^`Z}^]ªr�9ay�ãu`wy~�^¤��w1�9z{u`Z]\��{Z}|bZ}��_bz{^v�May_ba{z1�9\x��X¤Z4ux\`~��µ�G\`�9X[Z}��_b\`´�G_b\7��a1�9���Aä}~�ay�µuxwyZv~��yZ]a5ux~�®1�g�Lux~�_9a,_9���g���y^x�9��\`Z]����ux~�_bay^=wy~�zy^�~�a,�9Z]ayZ}\x�9����ay��g���y^x�9�YZ4­�Z}�}u`^H~�a�z1��\=ux~��]�y���9\]��åMZ7��~����Æ^`Z}Z[u`w1��u��]�9�y^x���¥�{~¦�9�b\`�9X[^ozy\`_�|A~��{Z�2zU_;��Z]\e�G�y�/X¤��uxw{Z]X¤��ux~��g�9�±ux_A_b�U~�a�uxw{~�^��9ay�9���A^`~�^]ÛAuxwyZ4���g��aµ¡UZP¢?�yZ]\=~�Z}�/ªy�y^O�~�a{�«Z4�Au`\`Z]X[Z]���M^=~�X[zy��ZFuxZ}^=u`^]ªYux_¸�{Z}uxZ}\`X[~�ayZ¤~���u`wyZv�9^=^`�yX[z{u`~�_bay^H�g|9��~��¦��¡y��Z�9\=ZP^`�{¶[�]~�Z]a5ur�G_b\�~��yZ}a5ux~��Q�A~�ay�2�g���y^x�9�°Z4­�Z}�}u`^]�¥sâ��^=_yªyuxwyZ��y~��9�b\`�9X¤^�zy\`_A�y�y�}ZX¤��uxwyZ}X��Lux~��]�9�±Z4��zy\=Z]^`^=~�_ba{^��G_9\��g�9�{^x�9�/Z4­�Z}�}u`^�~�a�uxZ}\`X[^�_9��_b¡y^=Z]\e|bZ]�µ�y~�^=ux\=~��¡y�{u`~�_ba{^]Û°_9u`wyZ]\e��~�^`Z�ª�uxw{Z2�y~¦���b\x�9X[^©�]�9aæ¡UZ2¢?�yZ]\=~�Z]�¸u`_v^`�y�9�bZ]^euS�9�y�y~�ux~�_9a1�9�_b¡y^=Z]\e|9�Lux~�_9ay^Æ_b\Y������~���~��9\=��Z���z¬Z}\`~�X[Z]a5u`^Æ�G\`_bX���wy~���w7u`wyZ��yZ}^`~�\=Z]�F~�a��GZ]\`Z}ay�]Z}^�g��a¨¡UZ�_b¡{u���~�ayZ}�/�

½©ay_�uxwyZ}\¯u`_�_9�©uxw1�Lu¯Z]X[Z]\`�9Z]^v�G\`_bX uxwyZ��b\`�9zywy~��g�9�o�9a1�9����^=~�^�_��H�]�9�y^x���Z}­¬Z]�4ux^o~�^o�æ@`>9C�@4ç�CEç5:¤6ON�IQK1Bt04ß4à904K1BDIJ6LK{:â»!�F^`Z4uP_9�Y~�a{�GZ]\=Z]ay�}Z2\`�{��Z]^ ¡5�µ��w{~���w^`Z}a5uxZ]a{�]Z]^©~�a5|b_b��|A~�ay�F~�a5u`Z]\=|9Z]a5ux~�_bay^ �9ay�¨_b¡y^`Z}\=|���u`~�_bay^��]�9a«¡UZPux\x��ay^=�G_b\=X¤Z}�~�a5u`_©_9u`wyZ]\$^`�y��w7^=Z]a5uxZ}ay�]Z}^]ª5uxw?�y^$zy\`_�|A~��y~�ay�o�r^=�Aa5u��9�4ux~���X¤Z4uxwy_A�H_��1�yZ]\=~�|A~�a{�cD_9\¯|9Z]\=~��Q�A~�ay�5jv�}�¦�9~�X¤^��9¡¬_9�{uµ~�a5uxZ}\=|bZ}a5ux~�_9ay^¸�9a{�!u`wyZ'���g�ÜuxwyZ4�Ü~�a5uxZ}\x�9�4u��~�u`w¾_b¡y^`Z}\=|���u`~�_bay^}��åÜ~�uxw�u`wyZ¯wyZ]��zÍ_9�ruxw{~�^7�g�����]�{���y^2u`wyZ¯\`Zg���yZ]\7��~�����¡UZ�9¡y��Zoux_«cD~¦j��yZ}u`Z]\=X¤~�ayZ�X���u`wyZ]X¤��u`~��g�������7��w{Z}uxw{Z]\ �F�9~�|bZ}a¨^=Z}u _9�$�]_�|��9\`~���uxZ}^~�^ �9zyz{\`_bzy\=~¦��u`ZS�G_9\ �}_ba5ux\`_9��_9�Y�}_ba{�G_b�{ay�y~�a{�yªÆcG~�~¦j��{Zg�9����~�u`w«X[Zg�9^=�y\`Z}X¤Z}a?u`^uxwy��uY��~�Z�_ba2u`wyZ��g�9�{^x�9�yzy��uxw5�����A^}ª{�9a{�«cG~�~�~Qj/ux\x���yZ�_bayZ�^=Z}u£_9�UX[Zg�9^=�y\`Z}X¤Z}a?u`^�G_b\��9a{_9uxwyZ}\]�

è�~�ay�9�����bª?��ZP��~�����^=wy_��ºwy_��×u`wyZ�ayZ}�×�g�9���]�y���y^��y~�^x�9XH¡y~��b�y��uxZ}^��]_ba{�]Z]z�ux^uxwy��u¥w1�g|bZ u`\`~��b�bZ]\=Z]�[�]_ba5u`\`_�|bZ}\`^=���9ay�¤X¤~�^`�}_bX¤XH�yay~��]��ux~�_ba7�9X[_bay��zyw{~���_9^`_��zywyZ}\`^}ª�^eu���u`~�^eux~��}~¦�9a{^]ª�Z}�]_ba{_bX¤~�^=u`^]ª��9ay��zy^e�A��wy_b��_9�b~�^eux^]��Þ�wyZ]^=Z¤~�ay�]���y�yZ2�{~�^e�ux~�ay�}u`~�_ba{^¥¡UZ}ut��Z]Z}av^=u`\`�y�4ux�y\`�9�U�9ay��\=Z]�b\=Z]^=^`~�_9a�Z]¢?�1�Lux~�_9ay^]ªA�yZ}®yay~�u`~�_bay^£_9���y~��\`Z}�}u£��ay�2~�a{�y~�\=Z]�}u�Z}­¬Z]�4ux^]ª?��ay�2\`Z}�¦��u`~�_ba{^`wy~�zy^$¡UZ}ut��Z]Z}a¤^eux\=�y�}u`�y\x�9��Z]¢?�1��u`~�_ba{^�9ay�¨é Z4�AX��9aA�Oê��y¡y~�avX[_A�yZ]��^]�

Page 3: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ�ëbÎÍÓOÖ7É$ËPÙ�Ò¯Ï�ÄrÉ�Ó4Ù�Ö d�ibìí¯î�ï ð�ñ�ò l£ó¯ôæõæq òYö ó ñÞ�wyZrzy\`_9¡y��Z}X¤^¥�9�y�{\`Z]^=^`Z}�v~�a7uxwy~�^£��w1�9z{u`Z]\��]�9a�¡UZ]^eu�¡UZ�~������y^eux\x�LuxZ]�7u`wy\`_b�{�bw���}�¦�9^=^`~��g�9�1Z��y��X¤zy��Z �{�yZru`_ · _A��wy\x��a�cD^=Z]Z åM�9~�ayZ]\�d�÷bf9÷5j4� · _bay^`~��yZ]\��9a�Z��?�zUZ]\`~�X¤Z}a5u°~�aP��wy~���wS^=_b~����G�yX¤~��5�9a5ux^ÆcGø'j/��\`ZY�y^=Z]��ux_�~�ay�}\`Z]�9^`Z£_5��u/�]\`_9z��A~�Z]���{^cÀùHj�¡5�F�]_ba5ux\=_b����~�ay� uxwyZ�Z]Z}����_b\`X*z¬_9zy�y�¦�Lux~�_9aµcJúoj�Û?u`wyZ��G�yX[~��5�9a5u`^ÆX��g�2�9��^=_w1�g|bZ¯�y~�\`Z}�}u7Z4­�Z}�}ux^µcG¡¬_�uxw�¡UZ]ayZ4®1�]~��9���9ay�¼�9��|bZ]\=^`Zgj7_9a,��~�Z]���y^2¡UZ]^`~��yZ�uxw{Z�]_9a?u`\`_b�±_9�/Z]Z]����_9\`X[^]��å�Zr��~�^`w[ux_2��^`^`Z}^`^�u`wyZ�ux_9ux�9�±Z}­¬Z]�}u�_9�/u`wyZ��G�yX¤~��5�9a5ux^_baÝ�A~�Z}���y^���wyZ}aáu`wy~�^�ut�Azy~��g�9�P^eux�y�{�+~�^v�]_9X¤zy��~��]��uxZ}�Ý¡?�Ü^=Z}|9Z]\x�����D���}ux_9\`^]�è�~�\`^=u]ª9�}_ba5ux\`_9����Z}�H\`�9ay�y_bX[~�ä}Z]�2Z���z¬Z}\`~�X[Z]a5u`^Æ�9\=Z��ya{�GZ]�9^`~�¡y��Z�»Ý�D�9\=X¤Z}\`^�~�ay^=~�^=u_ba,�{Z]�]~��y~�a{�¨�G_b\�uxwyZ}X¤^=Z]��|bZ]^2��w{~���w,zy��_9ux^��9\=Z¤ux_«¡¬Z[�G�yX¤~��5��u`Z]�/�¨¿�Z}�]_bay��ª�D�9\=X¤Z}\`^]û9��wy_b~��}Z�_9�¬u`\`Z]��uxX[Z]a5u¥�yZ]zUZ]a{�y^�_9a[���9^=uY�9Zg�9\}ûÂ^¥Z]Z}����_b\`X�z¬_bz{�y�¦��u`~�_bacJúYü�j�ª��9a¨�yay´?ay_���aæ¢?�1�9a5ux~�ut�¯uxw1��u ~�^�^eux\`_9ay�b���¯�}_b\`\=Z]����uxZ}�¨��~�u`w¨u`wy~�^��bZ]�9\]ûý^zU_bzy�y����ux~�_ba/�ÜÞ�w?�y^���Zæwy��|9Zæ�M�}�¦�9^=^`~��]�9�r�g�9^=Z¸_9���]_9a{�G_b�yay�{~�ay��¡y~¦�9^[uxw1�Lu~�a5u`Z]\=�GZ}\`Z}^H��~�u`wMuxwyZ��9^`^=Z]^`^=X¤Z}a5u¤_���ux\`Z]��uxX[Z]a5uHZ}­¬Z]�}u`^7\`Z}�5�9\`�{��Z]^=^2_9��^`�9XF�zy��Z[^=~�ä}Z9�¯è1_b\=u`�ya1��u`Z]���bª°uxwy\=_b�y�bw��¦�9¡U_b\`��ux_b\e�æ�9ay�9���A^`~�^�_9��^`_9~��Y^x��X¤zy��Z]^}ª���Z�g��a¤�yZ4uxZ}\`X[~�ayZ�uxw{ZrZ}Z]����_b\=X"z¬_9zy�y�¦�Lux~�_9ay^Æ¡UZ}�G_b\=Z©�9a{�����Qu`Z]\Æu`wyZru`\`Z]��uxX[Z]a5ugÛ�G�y\euxwyZ}\`X[_b\`Z�ªy¡¬Z}�g�9�{^`Z�u`wyZP�G�yX[~��5�9a5u`^��9\`Z�´?ay_���a¨u`_F¡¬ZS�9�4ux~�|9ZP�G_b\��[^=wy_b\euzUZ]\`~�_��[_bay���bª5��Z©�g��a�^x���GZ}���[�9^=^`�yX[Z�uxw1��u¥uxw{Z}�[�y_Ha{_9u��L­�Z}�}u�u`wyZr�b\`_���uxw¤_9�Z]Z}����_b\`X[^�^`�y\e|A~�|A~�ay�Fuxw{ZSu`\`Zg�LuxX[Z]a5ug�¥stay^=u`Zg�9��ª¬Z}Z]����_b\=X¤^}û1�b\`_���uxwµ�yZ}z¬Z}ay�y^_ba'uxwyZ¤z¬_bz{�y�¦��u`~�_ba�_9��¡y~�\=�y^�cD�9ay�,_�uxwyZ}\�zy\=Z]�1�Lux_b\=^�j4ª$��wy~���w�~�^S�]_b\=\`Z]����uxZ}���~�u`w����9^=u��9Zg�9\}ûÂ^$Z]Z]����_9\`XþzU_bzy�y����ux~�_ba��9ay��wyZ]ay�}Z���~�uxw�uxwyZ£ux\=Zg��u`X¤Z}a?u�~�ux^`Z}���O�

Þ�wyZ�X[Z}u`wy_A���yZ4|bZ}��_bzUZ]��~�a�uxwy~�^S��wy�9z{uxZ}\�zUZ]\=X¤~�ux^Puxw{Z[~�a5|bZ]^eux~��5��ux_9\�u`_ux\`�9ay^=�¦��u`Z��]_9X¤zy��Z4���]_9ay^`~��yZ]\`��ux~�_bay^�_��1uxwy~�^�^`_9\=uY~�a?u`_P� �G_b\=X����A���9ay�b�y�9�bZ���ay�uxw{Z]\`Z}¡5�v�D�9�]~���~�ux��uxZruxwyZP�G_b����_���~�a{�Hu��9^=´?^]ÿdb��Z��{z{��~��]��ux~�ay�2uxw{Z���^`^`�{X¤z{u`~�_ba{^�uxw1��ur�ya{�yZ]\=��~�Z uxwyZ�X[_��yZ}�JÛ� ���yZ}�]~��y~�ay�'��wyZ}u`wyZ]\�u`wyZæ�9^=^`�yX[z{u`~�_bay^��9\`Z¨^`�{¶[�}~�Z]a5u�u`_,_b¡{ux�9~�aÍ�]_baA�^=~�^=u`Z]a5u[Z]^eux~�X¤��u`Z]^F_9�ruxwyZ¯u��9\=�bZ}uF¢?�1��a?u`~�ut�b»Ýu`wyZ¯ux_9ux�9��Z}­¬Z]�4u�_��©uxw{Z�G�yX[~��b�9a5ux^�_ba¯�A~�Z}���y^}Û

Á{��zy\=_�|�~��y~�a{�¼cD~���u`wyZ¯�9ay^e��Z}\7ux_æ~�u`Z]X � ~�^7��¶[\=X���u`~�|9Z�jH�¸�}��_b^=Z]���J�G_b\`XZ��{z{\`Z]^=^`~�_ba×�G_b\vuxw{Z�u���\`�bZ4u¨¢?�1��a?u`~�ut�!~�aÜuxZ}\`X[^¨_��2�y~�^=ux\=~�¡y��ux~�_9ay^v_9�_b¡{^`Z]\e|bZ}�«¢?�1�9a5ux~�ux~�Z]^]Û±��ay�

Ãy��^=�y�b�bZ}^=ux~�ay��cG~���u`wyZ7�9a{^=��Z]\©u`_�~�u`Z]X � ~�^©ayZ}�5��u`~�|bZgj �[^`Z}uo_9�Y_b¡y^=Z]\e|9�;�u`~�_bay^r�9a{�¨Z��{zUZ]\=~�X[Z]a5ux^ruxw1�Lugª1~���z¬Z}\=�G_b\=X¤Z}�/ªy��_b�y���µ\`Z}ay�yZ}\o�F�]_bay^=~�^O�u`Z]a5urZ]^=u`~�X¤��uxZo�GZ]�9^`~�¡y��Z��

Page 4: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d�ibhyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�ÊÞ�wyZH®1\=^=uP^euxZ}zæ~�a¨u`wy~�^ �9a1�����A^`~�^©~�^�ux_��]_bay^eux\=�y�}u��[�g�9�{^x�9�$�y~¦�9�9\x�9X���~�´9Z

uxw{ZH_bayZS�b~�|bZ]a¨~�a«è�~��b�{\`ZSÁ{��dbª1��wy~���w¨\`Z]z{\`Z]^=Z]a5ux^ou`wyZH~�a5|bZ]^eux~��5��ux_9\]ûý^ �ya{�yZ]\O�

X

B

Z

Z

1Z

3

Y

0

Z2

è�~��b�y\`ZSÁ{��dbÿ�½*�]�9�y^x���$�y~��9�b\`�9X�\`Z}zy\`Z}^`Z}a?u`~�ay�¤uxwyZ�Z}­¬Z]�}u©_9���G�{X¤~��5�9a5ux^�cGø�j_bav�A~�Z}���y^�cÀùHj4�^=ux�9ay�y~�ay�¸_9��uxwyZ�X¤�;³=_b\S�g���y^x�9�¥~�a��1�yZ]a{�]Z]^F��X¤_ba{�«X[Zg�9^=�y\x�9¡{��Z[¢A�y�9a5ux~�u`~�Z}^~�aæu`wyZ¤�{_bX���~�a/�Fè1_b\PZ��y��X¤zy��Z9ª°uxw{Z[¢?�1��a?u`~�u`~�Z]^�ú����Hú����ú�H\=Z]zy\=Z]^`Z}a5uHuxw{ZZ]Z}����_b\`X zU_bzy�y����ux~�_ba¯¡¬Z4�G_b\`Z�ux\=Zg��u`X¤Z}a5ugª¬���QuxZ}\ru`\`Zg�LuxX[Z]a5ugª¬�9ay�æ�Lu u`wyZ�Z}ay�_9�°u`wyZP^`Z]�9^`_ba�ªy\`Z}^`zUZ]�}u`~�|9Z]���9�£Þ�wyZSú£ü�u`Z]\=X�\`Z}zy\`Z}^`Z}a?u`^��¦�9^eu��bZ]�9\]ûý^�Z]Z}����_b\`XzU_bzy�y����ux~�_ba/ÛL¡¬Z}�g�9�y^=Z�~�u$~�^$�9a2�{ay´?ay_���a7¢A�y�9a5ux~�ut�9ªb~�u$~�^��yZ}ay_9uxZ}�2¡5�2�rwy_b����_���]~�\`�}��Z9ª9�9^Æ~�^�u`wyZ�¢?�1�9a5ux~�ut� �¤ª9uxwyZ�z¬_bz{�y�¦��u`~�_baS_9�±¡y~�\`�y^��9ay�7_9u`wyZ]\$zy\`Z}�1��u`_b\`^}��°~�ay´?^�~�a�uxwyZP�{~¦�9�b\`�9X �9\`Z©_9��ut��_F´?~�ay�{^]ÿÆuxw{_b^`Zou`w1��u��}_bayayZ}�}u��yayX[Zg��^`�y\=Z]�¢?�1�9a5ux~�ux~�Z]^£�9\=Z��yZ]^=~��ba1�LuxZ]�[¡5�7�1�9^=wyZ]���9\`\=_���^]ª5uxwy_9^`Z��]_ba{ayZ]�4ux~�a{�SX[Zg��^`�y\=Z]�¢?�1�9a5ux~�ux~�Z]^P¡5�æ^`_9��~��¸�9\`\=_;��^}�7Þ�wyZF^`�y¡{^=u���a?u`~�|9Z��9^=^`�yX[z{u`~�_bay^PZ}X2¡U_��y~�Z]�æ~�auxw{Z©�y~��9�b\x��XÚ�9\`ZrayZ]�b��ux~�|bZ©�]�9�y^`�9����^`^`Z}\=u`~�_bay^¥��wy~���w¯��\`Zr�]_ba5|bZ4�bZ}�µu`wy\`_b�{�bwuxw{Z���~�ay´?^¤R7I�:�:4IQK��[�G\=_bX uxwyZ��y~¦�9�9\x�9Xµ�µè±_9\�Z��y��X¤zy��Z9ª�u`wyZ�X[~�^=^`~�ay�¸�9\=\`_��¡UZ}ut��Z]Z]a,ú��©�9a{��ù�^=~��ba{~�®1Z}^©uxwyZ7~�a5|bZ]^eux~��5��ux_9\]ûý^P�yay�yZ}\`^=ux�9ay�y~�ay�vuxw1�Lu�zy\`Z��ux\=Zg��u`X¤Z}a5u£Z]Z]����_9\`X[^Æ�g�9aFay_�u£��­¬Z]�}u£_b��uYzy���9a5ux^Y�y~�\`Z]�4ux���¬Ûbu`wyZ]~�\ÆZ]a5u`~�\`Z�~�a��1���Z]a{�]Z©_9a¤_5�Lu¥�A~�Z]���{^�~�^¥X[Z]�{~¦��u`Z]�F¡5�[u`wyZrz¬_b^eu`u`\`Zg�LuxX[Z]a5u��}_bay�y~�ux~�_bay^]ªAú�����ay�ú�g�¯²o�y\Szy�y\`zU_b^=Z¤~�^Sa{_9u�u`_µ|��9��~��1�LuxZ7_b\S\`Z}zy�y�y~���uxZF^`�{��w,�y_bX¤�9~�aA�t^`zUZ]�}~�®1��9^=^`�yX[z{ux~�_bay^�¡y�{u�\`��uxwyZ}\°ux_�uxZ}^=u���wyZ}u`wyZ]\$�r�9~�|bZ}a�^=Z}u�_9�y�9^`^=�yX¤z�ux~�_9ay^�~�^°^`���¦�®1�}~�Z]a5u��G_b\�¢A�y�9a5ux~��Q�A~�ay�7�g�9�{^x�9�/Z4­�Z}�}u`^r�G\=_bX�a{_bayZ4��zUZ]\=~�X[Z]a5u��9���1��u���»ÜwyZ}\`Z9ªZ]^eux~�X���u`~�ay��uxwyZPux_�u��9�/Z4­�Z}�}u _9���G�yX¤~��5�9a5ux^�_baµ�A~�Z]���{^]�

Þ�wyZP�g���y^x�9�¬�y~¦���b\x�9XÚ~�avè�~��9�y\`Z Á{��dP~�^�^=~�X[~����9\Y~�a�X¤�9a5�¤\`Z]^=z¬Z}�}u`^�ux_2uxw{Zz1��u`w¤�{~¦�9�b\`�9X[^Y�yZ}|A~�^=Z]�¤¡?�Få�\`~��bw5u ced�÷ � d;j4Û?¡U_9uxw[\`Z��1Z]�4u�u`wyZ ~�a5|bZ]^eux~��5��ux_9\]ûý^^`�{¡A³=Z]�4ux~�|9Z��9ay��¢?�1�9��~�u���u`~�|9ZY´Aa{_;����Z]�y�9Z�_9�1�]�9�y^`�9��~�a��1�yZ}ay�]Z}^�~�aSu`wyZ��y_9X��9~�a/ª¡U_9uxw�Z}X¤z{��_��F�y~�\=Z]�4uxZ]�¯���}�A�]��~��r�b\x�9z{wy^]ª{�9ay��¡U_9uxw��9����_;�ã�G_b\¥u`wyZo~�ay�]_b\=z¬_9\x�L�

Page 5: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ�ëbÎÍÓOÖ7É$ËPÙ�Ò¯Ï�ÄrÉ�Ó4Ù�Ö d�i��ux~�_baH_��1�¦��u`Z]a5uÆ_9\$�ya{X¤Z]�9^`�y\=Z]�7¢?�1�9a5u`~�ux~�Z]^}��Þ�wyZ�X��;³=_9\$�y~�­�Z}\`Z}ay�]Z}^Æ��~�Z¥~�aHuxw{ZX[Z}uxw{_��¨_9�£�9a1�����A^`~�^]�rè�~�\=^=u]ªU��wyZ}\`Z]�9^oz1��u`w«�y~��9�b\`�9X¤^�w1�g|9Z2¡¬Z}Z]a��9ay�9���Aä]Z}�X[_b^=u`���F~�a¤uxwyZ �]_ba5uxZ��Au�_9����~�ayZg�9\£X[_��{Z]��^£��~�u`w����9�y^`^=~¦�9a�ay_b~�^=Z9ª��g�9�y^`�9�U�y~¦�L��b\`�9X¤^rzUZ]\`X[~�u �9\=¡y~�u`\x�9\e�¯ay_bay��~�ayZ]�9\r~�a5uxZ}\x�9�4ux~�_9ay^]��sta«�D�9�4ugª¬_b�y\ �9a1�9����^=~�^r_9��g���y^x�9�°Z4­�Z}�}u`^���~����¬¡¬ZPZ}a?u`~�\=Z]����ay_bayzy�9\x�9X[Z}u`\`~��9ª{Z}a5u��9~���~�a{�2ay_F�]_bX[X[~�uxX[Z]a5uux_S�oz1��\=ux~��]�y���9\$�G�{ay�}u`~�_ba1���{�G_b\`X��G_b\YZ}¢A�y��ux~�_bay^£�9ay�F�y~�^=u`\`~�¡{�{ux~�_bay^]��¿�Z}�]_bay��ª�g���y^x�9�U�y~��9�b\x��X¤^Y��~����1¡¬Z �y^`Z}�vay_9u�_bay���[��^��Hzy�9^`^=~�|bZ©�¦�9ay�9�1�9�bZ�ux_��]_bX[X2���ay~��g��u`Z©�9^=^`�yX[z{u`~�_bay^¥¡y�{u��9��^=_H��^��9a��9�}u`~�|bZr�]_9X¤zy��u���u`~�_ba1���1�yZ}|A~��}Z u`wy\`_b�{�bw��wy~���w¸uxwyZ2�{Z]^`~�\`Z}�æ¢A�y�9a5ux~�u`~�Z}^o�9\=Z2�yZ]\=~�|9Z]�/��è1_b\ Z4�{�9X[zy��Z�ªUu`wyZ2X[Z}uxw{_��«u`_¡UZ��yZ]^=�]\`~�¡¬Z}��������_���^Y��a7~�a5|bZ}^=ux~��5��u`_b\Æux_P~�ay^`zUZ]�4uYuxwyZ��y~��9�b\`�9X*_��/è�~��b�y\=Z�Á{��d�9ay�µX¤�9´9Zou`wyZP�G_b����_;��~�ay�2~�X¤X[Z]�y~���uxZ �]_ba{�]���{^`~�_9ay^]�db��Þ�wyZ�u`_9u��9��Z}­¬Z]�4u�_��Uø�_ba[ùº�]�9aF¡¬Z�Z}^=ux~�X��LuxZ]�7�}_bay^=~�^=u`Z]a5ux���2�G\`_9X*uxw{Z_b¡{^`Z]\e|bZ}�«�y~�^=ux\=~�¡y��ux~�_9av_9�$ø��oú���� ú���oú�gª1��ay�«ù[�

� ��Þ�wyZPu`_9u��9�¬Z}­¬Z]�4u _���ø _baµù cD�9^`^=�yX¤~�ay�7�y~�^`�]\=Z}u`Z�|��9\=~¦�9¡{��Z]^¥uxwy\=_b�y�bwA�_b��u�j�~�^��9~�|bZ}a¨¡5�vu`wyZP�G_b\`X2�{�¦� �

�¤c������ j"! #%$'&(#)$�* #)$�+,��c-�.� /�0�1/2� j3�¤c4/��� /5��� j6 #87:9 ��c�/;�</5���1/�2� >= j3�¤c4/5�:� >= j�� cDÁ{��d�j

��wyZ}\`Z?�¤c������ j�^eu��9a{�y^��G_b\�u`wyZPzy\`_9¡1�9¡y~���~�ut�F_�������wy~�Z4|�~�ay�F�2�A~�Z]������Z4|bZ}�_9�Sù !@�¬ª��b~�|bZ}a+uxw1��u¤uxwyZ¨ux\`Z]��uxX[Z]a5uv~�^�^=Z}u�u`_,��Z4|bZ]� ø ! ¡?�Z���u`Z]\=a1�9��~�a5uxZ}\=|9Z]a5ux~�_ba/�Á{��½ �}_bay^=~�^=u`Z]a5uSZ]^eux~�X¤��u`~�_ba¨_9�£uxwyZ2u`_9u��9�$Z}­¬Z]�4u�_9�£ø _ba�ù���_9�y���æay_�u¡UZH�GZg��^`~�¡{��ZS~��¥ù���Z]\=Z7�]_ba{�G_9�yay�yZ}�¸��~�uxwæú��Û/w{_;��Z}|9Z]\}ª°�]_ba{�G_9�yay�y~�ay�ú��©��ay�µù"��~����¬ay_9ur~�a5|��9��~��1��u`Z uxw{Z��G_9\`X2�y���H�G_b\A�¤c������ j��

Þ�wyZ}^`Zµ�]_bay�}���y^=~�_ba{^H��~�����¡UZv_b¡{u���~�ayZ}��Z]~�u`wyZ]\H¡?���9a1�9����ä}~�ay�¸u`wyZv�b\`�9zywy~��g�9�zy\=_bz¬Z}\=u`~�Z]^µ_9��uxwyZ'�y~¦���b\x�9X _b\µ¡?�ÜzUZ]\e�G_b\`X[~�a{�Í�¼^`Z}¢A�{Z]ay�}Z,_9�2^e�AX2¡¬_9��~���yZ}\`~�|���u`~�_ba{^�cG�b_�|bZ}\`ayZ}��¡?�'uxw{Z¤�y~��9�b\`�9X�j©u`w1��u��b~�|9Z]^H\=~�^=Z[u`_«�g���y^x�9�¥Z}­¬Z]�4u�G_b\=X2�y�¦��^�^`�y��w«�9^�cÀÁ{��d;j��B'CEDGF�HGI�J�K�J4LMI�HON�P5QSRGTVUEK�WEX5W�FZY[L\H,]^D_K�`2J4F3a�b�c5LdJeLMWefgD_K�HGhiF3YjDGFkH5f'F'lSIia�J�D[J4IANmQ8Ronqpr TI�a^NmQSRsn t0u2Q r T�Twv5FZfkK�X5W�F�I�l5J�DGF.LMH_fkI�Hx�F3HGL\F3H_f'FELMHmD_K�H5Y2y\LMHGhzW4XGv_W�f'a�LM`2J�W3{�CED_F�a4FZK�Y2F3a>HGFkFZYHGI�J|v;FeLMH0J4LM}�LMYGK�J4FZYAL\l-~GK:J�J�DGL)W^`;I�LMH0J3~sQ��G{MbZTwK�`G`;F3K�agW^XGH2l8K�}�LMy\L)K�aZ{��Vl%J4F3aVa4FZK�Y2LMHGh��2F3f�J�L\IiH�G{ �_~�J4DGFVa�F3KiY2Fka>W4DGI�XGy)Y�v5F�K�vGy\FEJ�I�Y2Fka�L\xiFVW�X_fgD�lSI�a�}zX_yMKiW�UVLdJ�D�hia4FZK:J�Fka�F3KiW�F^J4D_K�HmW�Iiy\xLMHGhK�yMh�F3vGagK�L)fEF3�0X_K:J�L\IiH_W3{��.I�J4F�J4D_K�J r5� L)Ww}�Fka�FkyM��K�HmL\H5Y2F'�mI�lsW4XG}�}mK:J4LMI�H�J�D_K:J|a�K�HGh�FZW>I:x�F3aJ4D_F�x�K�y\X_F3WEI�l���{

Page 6: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d�ibfyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Êí¯î�� ð�ñ�ò mPl��¤m ñvòÆö ó ñ ögñ � pHl��Fó(� ö p ñ � ó¯ô¸mO�0��m�����3� ���>���O [¡¢��¡¤£ ¥A¦O§e¨'¡©¥�ª¬«s­�®�§���¯�§e­�®�°k¥�­[¡sta · w1�9z�uxZ]\�d�cJ¿�Z}�}u`~�_ba�d9� Á5j°��Z�^x�g�ãw{_;�¼�]�9�y^x���{X[_��yZ}��^}ª9�yay��~�´�Z�zy\=_b¡1�9¡y~���~�^e�ux~��PX¤_A�yZ}��^]ª��g�9aµ^`Z}\=|9ZSu`_[z{\`Z]�{~��}u�u`wyZ�Z}­¬Z]�4u©_9��~�a?u`Z]\e|bZ]a5u`~�_bay^}��Þ�w{~�^��9�y�{Z]��GZg�Lux�y\=Z7\`Z]¢?�y~�\`Z}^ouxw1��u©uxwyZ ³=_b~�a5uP�y~�^eux\=~�¡y�{u`~�_ba���¡UZ7^`�yz{zy��Z}X¤Z}a?u`Z]�æ��~�u`w���g���y^x�9���y~��9�b\`�9X7»+uxw1�Lu�~�^}ª¬�F�y~�\`Z]�4uxZ}�¸�9�4���}��~��S�9\x�9zyw�±�uxw1�Lu©~��yZ]a5ux~�®1Z]^�uxw{Z�g���y^x�9���]_baya{Z]�}u`~�_ba{^��9X¤_9ay��uxw{Z2|9��\`~¦��¡y��Z}^©_9�¥~�a5u`Z]\`Z}^=u]�Hsta¸uxw{~�^o^=Z]�}u`~�_baæ��ZZ]���9¡U_b\x��u`Z�_9a«uxwyZ�a1��u`�y\`Z�_9�£~�a5uxZ}\=|9Z]a5ux~�_bay^P�9a{�¸�b~�|9Z�Z���zy��~��]~�ur�G_b\=X2�y�¦��^��G_9\uxw{Z]~�\�Z4­�Z}�}u`^]�

Þ�wyZ¸�}_bayayZ}�}ux~�_baÝ¡UZ}ut��Z]Z]aÝu`wyZ¸�g���y^x�9�©�9ay�Ý��^`^`_A�]~���ux~�_ba1�9��\=Zg�9�{~�ay�b^¤_9�²�½1��^7~�^H�G_b\`X[Z]��u`wy\`_b�{�bw,uxwyZ¯X[Z]��w1��ay~�^=XF�â¡1�9^`Z}�¾�9�]�}_b�ya5u[_9�r�g�9�{^x��u`~�_ba/ª��wy~���w"_;��Z]^¾~�ux^M\=_�_�ux^�u`_ºZg��\`�����_b\`´?^�~�a�Z]�]_9ay_bX[Z}ux\=~��}^ÝcÀè1\=~�^`��w�dg÷bÁbf{Û³©�9��|9Z]��X¤_µd�÷9Ã5Á�Û±¿�~�X[_ba�d�÷5ì�Á5j4�¥staµu`wy~�^r�9�}�]_b�{a?u]ª���^`^`Z}\=u`~�_bay^r�9¡U_b�{ur�]�9�y^x���~�a��1�yZ]a{�]Z]^}ª�^=�y��wæ�9^�uxwy_9^`ZH^=z¬Z}�]~�®1Z]�¸¡5�vu`wyZH��~�ay´?^�~�a¨è�~��9�y\`ZSÁ{��dbª±^=ux�9ay�¨�G_9\�9�{u`_bay_bX[_b�y^¥zyw5�A^`~��g�9�¬X¤Z}��w1�9ay~�^`X[^��9X[_bay�SuxwyZ �]_b\=\`Z}^`zU_bay�y~�ay�2¢?�1�9a5ux~�ux~�Z]^]Ûuxw{Z]^`Z�X[Z]��w1��ay~�^=X¤^Y��\`Z�\`Z}zy\`Z}^`Z}a?u`Z]���9^Æ�G�yay�4ux~�_9a1�9�{\=Z]����ux~�_bay^`w{~�zy^$z¬Z}\=ux�{\`¡UZ]�¡5�º\`�9ay�y_bX �y~�^=ux�{\`¡1�9a{�]Z]^}�¹WY��wy_9~�ay�+u`wy~�^µux\x���y~�u`~�_ba/ª?´�Zg�9\=�7�9ay�¶µ£Z}\`X¤�cedg÷b÷{d;j�~�a5uxZ]\=zy\`Z4uxZ}�µu`wyZP�g�9�{^x�9�°\=Zg�9�y~�ay�7_9����²r½1�*~�a�uxZ}\`X[^�_9���G�yay�}u`~�_bay�9�Jª\x�LuxwyZ}\$uxwy�9aHzy\=_b¡1�9¡{~���~�^=u`~��9ª;\=Z]�¦�Lux~�_9ay^`wy~�zy^�cD^=Z]ZPcOdb�ýÃ5i5j��9ay�·²oZ4®1ay~�u`~�_ba � � � � � j4Û~�a«_9uxwyZ}\©��_b\`�y^}ª°Zg�9��wæ��w{~����A�tz1�9\=Z]a5u©�D��X¤~����¯~�a¸�¸²r½1�¹± \=Z]zy\=Z]^=Z]a5ux^S���yZ��uxZ}\`X[~�ay~�^=u`~�� �G�{ay�}u`~�_ba

>º !¶» º cS¼w½ º �:¾ º j�� ¿e!�d��À8À�À8��ÁÂ� cDÁ{� � j��wyZ}\`Z¸¼w½ º �9\=Z«uxwyZæzy�9\`Z}a?u`^v_9��|��9\`~��9¡y��Z¯ø º ~�añFÛouxw{Zľ º cedÃÅÆ¿ÇÅ@Á/j�9\=Z�X2�{u`�1�9������~�ay�{Z]zUZ]ay�yZ}a5ugª?�9\=¡y~�u`\x�9\=~����o�y~�^eux\`~�¡y�{u`Z]�7\x��ay�y_bX�y~�^eux�y\=¡1�9ay�}Z]^}�Þ�wyZ}^`Z¥�y~�^eux�y\=¡1�9ay�}ZÆuxZ]\=X¤^°\=Z]zy\=Z]^=Z]a5u�~�ay�yZ}z¬Z}ay�yZ}a?u�¡1����´A�9\`_b�ya{�S�D�9�4ux_b\=^/uxw1�Luuxw{ZH~�a5|9Z]^=u`~��5�Lux_b\r��wy_A_b^`Z}^oay_9u ux_�~�ay�]���y�yZS~�aµuxwyZH�9a1�9����^=~�^}�rsâ�¥��a?�µ_9�Yu`wyZ]^=Z�D�9�4ux_b\=^¥~�^¬³=�y�y�bZ}�[u`_�¡UZ�~�a��1�yZ]a{�]~�a{�Sut��_�_9\¥X[_b\=Z�|9��\`~¦��¡y��Z}^�cGu`wA�{^¥|A~�_9�¦��u`~�ay�uxw{Z�~�ay�yZ}z¬Z}ay�yZ}ay�]Z��9^=^`�yX[z{u`~�_ba±j�ª9uxw{Z]a7uxw1�LuÆ�D�9�}u`_b\ÆX2�{^=uYZ}a?u`Z]\Æu`wyZ��9a1�9���A^`~�^�9^��9aµ�yayX[Zg��^`�y\=Z]�McD_b\�����uxZ}a5u�j�|��9\`~��9¡y��ZP�9ay�¯¡UZP\`Z]z{\`Z]^=Z]a5uxZ}�¸~�a�u`wyZ��b\x��zyw¡5�'�¨wy_9����_��*ay_A�yZ9ª�^`�y��w,�9^2úYü2��ay�È��~�a�è�~��9�y\`ZFÁ{��db��è±_9\SZ��{�9X¤z{��Z9ª�uxw{Z�g���y^x�9���9^`^=�yX[z{ux~�_bay^��}_ba5|bZ}�9Z]�¨¡?�¤uxwyZ©X¤_A�yZ]�U~�a�è�~��b�{\`ZoÁ���dP�}_b\`\=Z]^`zU_bay��u`_

Page 7: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ\É{ÎÍÓOÖ7É$Ê£ËAÊ�Ê£ÖFÉ�Ó4Ù�ÖÓOÖըǩËjËHÙ?Ê$ÓtÇoÖ*Õ,Ù�ÒPÊ¥Ñ�Ð d�i9÷uxw{Z��G_9����_���~�ay�2^`Z4ur_9��Z}¢?�1��ux~�_bay^}ÿ

úYü@!Ì»Lü;c�¾`ü4j�� � !�»5Í¥cJúYü�:¾ZÍ$j��ú��Î!Ì»o�}cÀú£ü��¾��`j�� ø !�»GÏPcJúYü�:¾3Ï�j��ú��@!Ì»_�;cQøÇ��ú����:¾Z�4j�� ù !�»_Ð�cGø���ú����ú��:¾ZÐ$j��ú�@!Ì»_;c��Ñ��ú���:¾Z4j�À

cDÁ{� Ábj

§¨_b\=Z��bZ}ayZ]\`�9�����bª���Z�X¤�g�����yX[z�u`_b�bZ4uxwyZ}\H�9���£�yay_b¡y^=Z]\e|bZ]���D�9�4ux_b\=^�cG~�a���]���y�y~�ay�[u`wyZ ¾ º j�~�a5ux_��[^`Z4u,Ò*_9�Y¡1����´A�9\`_b�ya{�¸|9��\`~¦��¡y��Z}^ ��ay�¨u`wyZ]a¸^=�yX[X��L�\`~�ä]Z2u`wyZ]~�\P��w1�9\x���}uxZ}\`~�^=ux~��]^P¡5���v�y~�^=ux\=~�¡y��ux~�_9a¸�G�yay�}u`~�_baÓ�¤c�Ô¬jâ»Ý_b\©¡5�æ^=_bX[Z�9^=z¬Z}�}ux^µcGZ9�ý�y�Æ~�ay�yZ]zUZ]a{�yZ]ay�}~�Z}^�jH_9�1�¤c�Ô¬j4�æÞ�w?�y^}ª��«�G�{���¥^`zUZ]�]~�®1�g�Lux~�_9a�_9�r��g���y^x�9��X[_A�yZ]����_b�y���¼Z}a5u��9~���ut��_'�]_bX[zU_bayZ]a5u`^]ÿ¨��^=Z}u¤_9�r�G�yay�}u`~�_bay�9��\`Z}�¦�L�ux~�_bay^=wy~�zy^

>º !Ì» º cS¼w½ º ��Ô º j�� ¿e!*d;�ÀÀÀi��ÁÂ� cDÁ{�ÂÃ5j�9ay�¾�7³=_b~�a5u2�{~�^=u`\`~�¡y�{ux~�_ba��G�{ay�}u`~�_baÕ�¤c�Ô¬jP_ba,uxw{Z�¡1�9��´?�b\=_b�yay���D�9�4ux_b\=^]�æsâ�uxw{Z7�y~¦�9�9\x�9XÖ±¤c�׺jo�9^=^`_A�]~���uxZ}�¸��~�uxw����]�9�y^`�9�ÆX[_��yZ}��× ~�^©�9�}�A�}��~���ª�uxw{Z]a× ~�^ �]�9����Z]�æ:�04R7IÙØ�Ú�>9ß�Û56Là�ID>9K1��sâ�OªU~�a«�9�y�y~�ux~�_9a/ªyuxwyZ�¡1�9��´?�b\=_b�yay�«|��9\=~¦�9¡{��Z]^�9\=Z¤~�ay�yZ]zUZ]a{�yZ]a5ugª�× ~�^��g������Z}�ÜÚæ>9ß�Û56Là�ID>9K1ª$^=~�ay�}Z¤u`wyZ¤\=Z]^=�y��u`~�ay�µ�y~�^eux\=~�¡y���ux~�_bav_9��uxwyZ�_b¡{^`Z]\e|bZ}�¨|��9\`~��9¡y��Z]^���_b�y���¯u`wyZ]aµ¡UZS§«�9\`´�_�|¯\`Z]����ux~�|bZ©u`_ݱ�c4׺jcD^=Z]Z�Þ�wyZ]_b\=Z]X db�ýÃy��d�j4��Þ�w?�y^}ª¬uxw{ZHX[_��yZ}�°�yZ]^=�]\=~�¡UZ]�¸~�a¸è�~��b�y\`ZSÁ{��dH~�^�^`Z]X[~��§«�9\=´9_�|A~¦�9a�~���uxwyZ¥_b¡y^=Z]\=|9Z]�2|��9\=~¦�9¡y��Z]^��9\=Z1Þ�ø���ùz��ú�����ú��i��ú�iß?Ûb~�u���_b�y���Sux�y\=a§«�9\=´9_�|A~¦�9aF~��¬ú£ü£�9a{���×��Z}\`Z�_b¡y^=Z]\=|9Z]�¤�9^���Z]���J��sta · wy�9z{uxZ}\z� ��Z���~����Azy�y\O�^`�{Z�uxw{Z2�9a1�����A^`~�^ _9�Æ�bZ]ayZ}\x�9��ay_baA�e§«�9\=´9_�|A~¦�9a«X¤_A�yZ]��^]ª1¡y��uo~�aµuxwy~�^ ��wy�9z{uxZ}\�9���/X¤_A�yZ}��^��9\=ZS�9^`^=�yX¤Z}�µu`_¤¡UZ�Z]~�uxwyZ}\r§«�9\`´�_�|�~��9aµ_b\ §«�9\`´�_�|�~��9a¯��~�uxw¯�yaA�_b¡y^=Z]\e|bZ]�¨|9��\`~¦��¡y��Z}^�cD~À�ÂZ��y^`Z}X¤~ �e§«�9\=´9_�|A~¦�9a1j4�

é Z}Z]�y��Z]^`^¥ux_S^=ux��uxZ�ª?��Z ��_b�y���¤^=Z]���y_bX ¡UZ�~�a[z¬_b^=^`Z}^`^`~�_ba�_��V�¤c�Ô¬j$_b\£Z}|9Z]a» º � sâu©~�^r~�X[z¬_9\=u���a?urayZ4|bZ}\=uxw{Z]��Z}^`^ou`_�Z4��zy��~��]��uxZ�uxwyZ�X��LuxwyZ}X���u`~��]�9�°�]_ba5u`Z]a5u_9�±���G�y�����o^=z¬Z}�]~�®yZ]�2X[_��{Z]�?~�aS_b\`�{Z]\$u`_ �{\x�g��|��9��~��S~�a{�GZ]\=Z]ay�}Z]^��G\=_bXþz1�9\eux~��9�����^`zUZ]�}~�®1Z}�¨X[_A�yZ]��^]ªy^=�y��w¸�9^�u`wyZ�_bayZ��yZ]^=�]\=~�¡UZ]�µ~�aµè�~��9�y\`ZPÁ���db�

Þ�wyZ7Z]¢?�1�Lux~�_9a1�9��X¤_A�yZ]��cÀÁ{� � j�~�^ u`wyZ2ay_9ayz1�9\`�9X¤Z4ux\=~��2��a1�9��_9��_9�Yu`wyZ2^`_L��g������Z}�v^=u`\`�y�4ux�y\`�9�°Z]¢?�1��u`~�_ba{^rX[_A�yZ]�£cQåM\=~��bw5uPdg÷ � dbÛw�P_b���y¡¬Z}\`�bZ}\od�÷o��Á5j4ªyZ��?��]Z}z{uSuxw1��u]ÿouxwyZF�G�yay�4ux~�_ba1�9���G_9\`X�_9�¥uxw{Z[Z}¢?�1��ux~�_bay^¤cÀ��^���Z]���£�9^PuxwyZ[�y~�^=ux\=~��¡y�{u`~�_ba�_9��uxwyZP�y~�^=u`�y\`¡1��ay�]Z©u`Z]\=X¤^xj¥��~����¬\=Z]X¤�9~�a��{ay^`zUZ]�}~�®1Z}�/�£Þ�wyZPZ]¢?�1�9��~�ut�^`~��bay^7~�a�^=ux\=�y�}u`�y\x����Z]¢?�1��u`~�_bay^7�}_ba5|bZ4��uxwyZ¯�9^e�AX¤X[Z}u`\`~��]�9���}_b�ya5uxZ}\=�D�9�4ux�1���\`Z}�¦��u`~�_ba'_9�Ýàe~�^S�yZ}u`Z]\`X[~�a{Z]�,¡5�bªdá'�9ay�,Z]�9��w�Z]¢?�1�Lux~�_9aM\`Z}zy\`Z}^`Z}a?u`^[�¨^=ux�9¡y��Z�9�{u`_bay_bX[_b�y^rX[Z]��w1�9a{~�^`Xµ��è1_b\rZ4�{�9X[zy��Z�ª1uxwyZ�Z]¢?�1��u`~�_baµ�G_b\rù�^=ux��uxZ}^ u`w1��ugª\`Z}�5�9\=�y��Z}^`^�_9����w1��u���Z��]�y\=\`Z]a5u`����_9¡y^`Z}\=|bZ��9¡U_b�{urù"�9a{�µ\=Z]�5��\`�y��Z]^`^�_�����a?�

Page 8: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

dbd�iyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê��w1�9a{�bZ]^£uxw1�Lu£X¤~��bw5u£_��}�]�y\£~�a7_�uxwyZ}\¥Z}¢?�1��ux~�_bay^}ªA~��¬|��9\=~¦�9¡{��Z]^�cGøÇ��ú����ú���¾ZÐ$j��Z]\`Z�ux_o��^`^`�{X¤Z¥uxwyZ¥|9�����yZ}^�c �:/�2�:/�:¾3Ð�j�ª9\=Z]^`zUZ]�4ux~�|bZ]���bª9u`wyZ]a2ù!��_b�y���Su��9´�Z_bavuxw{Z�|��9���yZP�y~��}u��LuxZ]�µ¡5��uxw{Z��G�{ay�}u`~�_baâ»_У�

êrZ}�g�9����~�ay�H_b�{\��y~�^=�]�y^=^`~�_ba¯~�a¯¿�Z}�}u`~�_baædb�ýÃyªAuxwyZo�G�ya{�}ux~�_ba1�9�U��w1�9\`�9�}u`Z]\=~�äg�;�ux~�_ba�_9�yZ]�9��w2��wy~������âz1�9\=Z]a5u�\`Z]����ux~�_bay^=wy~�zP��Zg�9�y^°u`_ru`wyZ�^`�9X[Z�\=Z]�]�{\`^`~�|bZ��yZ]�}_bXF�zU_b^`~�ux~�_9a�_9��uxw{Z¥³=_b~�a5u��{~�^=u`\`~�¡y�{ux~�_ba¤u`w1��u���wy�9\x�9�4uxZ}\`~�ä}Z]^Aã��g�9Z]^`~��9a¯ayZ}ut��_b\=´A^}ÿ

��c ���À8À�À8� >ä jz!�å º �¤c >º �2¼w½ º j�� cDÁ{� ì9j

��wy~���w/ª1~�a¯_b�y\�Z��{�9X¤z{��ZP_9�Æè�~��b�{\`ZPÁ{��dbª{�A~�Z]���{^�¤c4/gü0� �:/5���:æ2�:/�2��/���{jç! ��c�/gü�j3�¤c � /gü�j3�¤c4/5��</gü4j3�¤c4æ�� /gü�j

6 �¤c4/��� �:/5�=jk��c�/;�</�2�:æ4jk��c-�.� �:/�0�:/�j�ÀcDÁ{� hbj§¨_b\=Z]_�|bZ}\]ªruxwyZ«�G�{ay�}u`~�_ba1��� ��w1��\x�9�4uxZ]\=~�ä]��ux~�_baãzy\=_�|�~��yZ]^¯���]_ba5|bZ}ay~�Z}a5u¯�¦�9a���b�1���bZ¸�G_b\�^=z¬Z}�]~��Q�A~�ay��wy_���u`wyZæ\`Z}^`�y��ux~�a{���y~�^eux\`~�¡y�{u`~�_baã��_b�y���Ü��w1�9ay�bZ�~�a\`Z}^`zU_bay^=Z¤ux_¨Z4�AuxZ}\`a1����~�a?u`Z]\e|bZ]a5u`~�_bay^}�vÞ�wy~�^�~�^H�9�}�]_bX[zy��~�^`w{Z]�'¡?��Z}ay�]_A�y~�ay�Zg����w«~�a5u`Z]\=|9Z]a5ux~�_ba¨�9^r�9a«����uxZ}\x��u`~�_ba�_ba«�F^`Z}��Z}�}ur^`�y¡{^`Z}u©_9���G�yay�}u`~�_ba{^���w{~���Z´9Z}Z]zy~�ay�¯uxwyZF_9u`wyZ]\P�G�yay�4ux~�_bay^P~�a5ux�9�}u]�[²Pa{�]ZF��ZF´?ay_�� uxw{Z[~��yZ]a5u`~�ut�¸_���uxw{ZX[Z]��w1�9a{~�^`X[^��9��u`Z]\=Z]��¡5�PuxwyZ¥~�a5uxZ}\=|9Z]a5ux~�_baH��ay�Su`wyZ¥a1��ux�{\`Z£_9�{u`wyZ¥�9��u`Z]\`��ux~�_ba/ªuxw{Zo_�|bZ]\`�9���UZ}­¬Z]�4u�_9�/u`wyZo~�a?u`Z]\e|bZ]a5u`~�_ba��]�9av¡UZ©zy\=Z]�y~��}uxZ}�v¡5��X[_��{~��Q�A~�ay��uxw{Z�]_9\`\`Z}^`zU_bay�y~�ay�2Z}¢A�y��ux~�_bay^�~�a¤uxw{ZoX¤_A�yZ}���9a{���y^`~�ay�Hu`wyZoX[_��y~�®1Z]�¤X¤_A�yZ}�±u`_�]_9X¤zy��uxZS�7ayZ}�×zy\`_b¡y�9¡y~���~�ut�[�G�yay�4ux~�_9a/�

Þ�wyZ7^`~�X¤z{��Z]^eu ut�AzUZ7_9�£Z4�Au`Z]\`ay�9�Æ~�a5u`Z]\=|9Z]a5ux~�_baæ~�^ _bayZ2~�a«��wy~���w���^`~�ay�b��Z|��9\`~��9¡y��Z9ª�^`�g�Íø º ª�~�^¤�G_b\=�]Z}�ãux_'u��9´�Z¸_ba+^=_bX¤Zµ®{��Z]�+|��9���yZ >º �º¿��{��wÜ�9a~�a5u`Z]\=|9Z]a5ux~�_ba/ªÆ��wy~���wM��Zv�g�9���èà=��ux_bX[~���ª\á¸�9X¤_9�ya5ux^�u`_¸��~��Qux~�ay�vø º �G\`_bX uxw{Z~�a��1�yZ]a{�]Z7_9�£uxw{Z7_b���«�G�{ay�}u`~�_ba1����X[Z]��w1�9a{~�^`X >º !é» º cS¼>½ º ��Ô º j©��ay��zy�¦�9�}~�ay�~�u��yay�yZ}\�u`wyZ�~�a��y�yZ]ay�}Z�_9�Æ�7ayZ4�ºX[Z]��w1��ay~�^=X�uxw1�Lu�^`Z}u`^ru`wyZo|��9���{Z >º ��w{~���Z´9Z}Z]zy~�ay���9���A_�uxwyZ}\�X[Z]��w1�9a{~�^`X[^��yayzUZ]\=u`�y\`¡UZ]���Yè1_b\`X¤�9�����bª�uxw{~�^Æ�Lux_bX[~��£~�a5u`Z]\e�|bZ}a5ux~�_9a/ªA��w{~���w¤��Z©�{Z]ay_9u`Z©¡5�Ýêsë{cQø º ! >º j4ª?_b\�êsë{c >º jÆ�G_b\£^`wy_9\=ugª � �9X[_b�ya5u`^ì3��H�F3�0XGLMx�K�y\F3H0J�HGI�J�K:J�L\IiHs~:X_W4L\H_hÂíZîZï'Q r T�LMH_W-J�F3KiY�I�l_t0u2Q r T�~�UEK�W�X_W4F3Y�LMHzð�FZK�a�y5Q-bZñ�ñ0ò�KiT'{CEDGF�tiu2Q r T^HGI�J�K:J�L\IiHAUEK�W�ó_a�W�J^X5W�FZYALMHõôÂI�y)YGW4ök}�L)YJ|K�H_Y1ð�FZK�a�y�Q-bZñ�ñ0÷�T>K�H_YAL)W�h0K�LMHGLMHGhzLMH`;I�`GXGy)K�aEW4XG`G`;I�a4J3{>CEDGF.Fk�2`Ga4FZW4W4LMI�HANmQSRsn tiu2Q r T4T�L)W^F3�0XGLMx�K�y\F3H0J^LMH1LMHiJ�FkH0J�J�I�NmQ�ø Pzù RGTwLMHJ4D_F�`5I�J4F3HiJ�LMK�ydúûI�XGJ�f'Ii}�FÂ}�I2Y2F3y�LMHiJ�a4I2Y2X5f'F3Yõv�j�.Fk�}mK�H·Q-bZñi÷��iT|K�H_Y[üVX_vGL\H�Q�b3ñý��0T|K�H_YJ4IjJ4D_F�Fk�`_a4FZW4W4L\IiH?NmþMQS� ù r T�ÿ�� Q8ø ù RGT��>L\H?J4DGFmfkI�XGH0J4F3a�l8Kif�J�X_K�y�J4DGF3I�a��OI�l��sF3UVLMWQ-bZñ0ý���v5T'{zCEDGFmW�F3}mK�H0J4L)fkK�y�Y2L���Fka�FkH_fkF3W�K�}�IiHGhAJ�DGF3W4F�HGI�J�L\IiH_W�K�a�F�Y2L)W�f'X_W�W�FZY(LMH �2F3f�J�L\IiH�G{ �G{ ��K�H_YõLMH(]^D5K�`2J�Fkaeý{

Page 9: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ\É{ÎÍÓOÖ7É$Ê£ËAÊ�Ê£ÖFÉ�Ó4Ù�ÖÓOÖըǩËjËHÙ?Ê$ÓtÇoÖ*Õ,Ù�ÒPÊ¥Ñ�Ð dbd9dux_P\=Z]X[_�|�~�ay�ouxw{Z�Z]¢?�1��u`~�_ba >º !¶» º cS¼>½ º ��Ô º j��G\=_bX*u`wyZ�X[_��yZ}�{��ay�F^`�y¡y^eux~�ux�{ue�~�a{��ø º ! >º ~�a¸uxwyZH\`Z]X¤�9~�ay~�ay�¤Z]¢?�1��u`~�_ba{^]��Þ�wyZ7ayZ4��X[_��{Z]�$u`w?�y^P�]\=Zg��u`Z]�\`Z}zy\`Z}^`Z}a?u`^¤u`wyZ¯^=�A^=u`Z]XµûÂ^F¡UZ]w1�g|A~�_b\F�ya{�yZ]\7uxw{Zµ~�a5uxZ]\e|bZ}a?u`~�_ba©êsë�cGø º ! >º j�9ay��ªU��wyZ}aæ^`_b��|bZ}�¸�G_b\ru`wyZH�{~�^=u`\`~�¡y�{ux~�_baµ_9�ÆøLª±�A~�Z}���y^�uxwyZ��g�9�{^x�9�$Z}­¬Z]�}u©_9�ø º _baæø�;ª°��wy~���w'~�^P�{Z]ay_9u`Z]� ��c s�4� >º j���§µ_b\`ZF�bZ}ayZ]\`�9�����bª���wyZ}a,�9a'~�a5u`Z]\e�|bZ}a5ux~�_9a��G_9\`�]Z}^P��^`�{¡y^`Z4uSø _9�£|9��\`~¦��¡y��Z}^ ux_���u`ux�9~�a«®��{Z}�æ|9�����yZ}^ ª/uxwyZ}a��^`�{¡y^`Z4u�_9��Z]¢?�1�Lux~�_9ay^�~�^/ux_�¡UZYzy\`�{ayZ]�S�G\`_bX!uxwyZ£X[_��yZ}�b�b~�|bZ]a�~�a�cDÁ{�ýÃ?j4ª;_bayZÆ�G_9\Zg����w�X[Z]X2¡UZ]\Y_9�¬ø�ª5uxw?�y^¥�yZ}®yay~�ay����ayZ}�Ý�{~�^=u`\`~�¡y�{ux~�_ba7_�|bZ}\¥u`wyZ�\`Z]X¤�9~�ay~�ay�|��9\`~��9¡y��Z]^�u`w1��ur�]_bX[zy��Z}uxZ}���¤��w1�9\`�9�}u`Z]\`~�ä]Z}^ru`wyZ�Z}­¬Z]�}ur_9��uxw{ZS~�a5uxZ]\e|bZ}a?u`~�_ba�� �� ����������������������� "!$#%'&)(*%'+-,/.0 213�345 IQà904K�B�6�6¨T9I�:�7g6LIQK1B�:�04BG:v6ONFàL>9ß�ID>58]C�0�:98Yø >9KUT¤ù:8 BG.10��]�9�y^`�9�£Z}­¬Z]�}u¨6ONoø6LK�ù;8�TA04K�6LBt0`T,04IQBQ.104ßv>L: �¤c������ jµ6L߯>L: �¤c����<êsë{c j`j<8oI�:v>7N�çAK�@4BDIJ6LKµN�ßx6LRø Bt6µBG.107:>={>A@�0�6ON?=Uß�698�>58]IQCEIQBD3«T9I�:4BÀß�ID8]çABÀIJ6LK{:�6LK�ù�VA@$6Lß�0`>A@�.�ßx0`>9CEICB�>9BDIJ6LK 6ON�ø��O��c-�.�4� j1�9IQà90�:�BG.10D=Uß�698�>58]IQCEIQBD3v6ON�ù ! ��IQKUT9ç±@�0`T�8]3�TA04C�04BDIQK���N�ßx6LRBQ.10µR�6]TA04Cr6ON¨cÀÁ��ÂÃ?jµ>9CQC 0FE]ç{>9BÀIJ6LK{:«@�6Lß4ßx0�:>=16LKUT9IQK���BO6'àL>9ß�ID>58]C�0�:vIQK�ø >9KUT:4ç{84:4BÀIQBDç�BDIQK��Hø ! IQK¸BQ.10Sß�04R[>9IQK1IQK��v0FE]çy>9BDIJ6LK{:]V· ��Z]�9\`���bªruxwyZ��b\`�9zywÜ�]_b\=\`Z}^`zU_bay�y~�ay�¼u`_�uxwyZ'\`Z}�y�y�]Z}�×^=Z}uµ_9��Z]¢?�1��u`~�_bay^¯~�^�¯^`�y¡y�9\x�9zyw�_9��±��G\`_bX���w{~���w,�����Y�9\=\`_���^SZ]a5uxZ}\`~�a{�µø w1�g|bZ[¡¬Z}Z]a�zy\`�ya{Z]�cJ¿Azy~�\euxZ]^�Z}uÆ�9�J�5dg÷b÷bÁ5j��$Þ�wyZ��y~�­¬Z]\=Z]ay�}Z-G�cDù¬� ê�ë�c = j`jIH�G�cÀùâ� ê�ë�c = = j`j�~�^�^`_bX[Z4�ux~�X¤Z}^�ux�9´9Z}a¯�9^�uxw{Zo�yZ}®yay~�u`~�_ba¤_9�jà=�g���y^x�9�¬Z}­¬Z]�4u�áµcÀê�_b^=Z]a?¡1�9�yX��9ay�¯ê��y¡y~�ad�÷bf9Á5j4ªY��wyZ}\`Z = ��ay� = = ��\`Zvut��_'�y~�^=ux~�ay�}uF\=Zg�9��~�ä]��ux~�_bay^H_�� ø��¾Þ�wy~�^7�y~��¦��GZ]\=Z]ay�}Z��g�9a¨�9�����g�A^�¡¬ZP�}_bX¤z{�{uxZ}�µ�G\=_bX u`wyZ��bZ]a{Z]\x�����G�yay�4ux~�_9aâ�¤c����<êsë�c j=j4ª��wy~���w+~�^F�yZ4®1ayZ]�+�G_9\¤Z4|bZ]\e�¼��Z}|9Z]� _9�©ø �9ay�+z{\`_�|A~��yZ}^��'X¤_b\=Z¨\=Z}®1a{Z]���w1�9\`�9�}u`Z]\=~�äg�Lux~�_9aµ_��$u`wyZ�Z}­¬Z]�}ur_9�$~�a5u`Z]\=|9Z]a5ux~�_bay^}�

�m������� «s­�®�§���¯�§e­�®�°k¥�­[¡Ã��¡KJÓ���w°'�0LO¨k§�¡½©a¤�9��uxZ]\=a1��u`~�|bZPcD¡y�{u¥^=_bX¤Z4ux~�X¤Z}^ÆX¤_9\`Z��9zyzUZg����~�a{�?j$�9�}�]_b�{a?u�_9�¬~�a5uxZ}\=|9Z]a5ux~�_baux\=Zg��u`^¥u`wyZ��G_b\`�}Z \=Z]^=z¬_ba{^`~�¡{��Z��G_b\Æuxw{Z ~�a?u`Z]\e|bZ]a5u`~�_ba��9^¥��|��9\=~¦�9¡y��Z���~�u`wy~�aFuxw{ZM3CEDGFÂv_K�W4LMf�x0LMFkU I�lsL\H0J�Fka�x�FkH0J�L\IiH_W�KiW�FZ�iX5K:J4LMI�Hm}�I2Y2L\ó_FkagW^I�a�L\hiL\H_K�J4FZW>UVLdJ�DON(K�a�W�fgD_KQPQ-bZñiòQRiTEK�H5Y?�LM}�IiH�Q-bZñiò��0T�{��.H(F'�2`GyMLMfkLdJeJ4agK�H_W4y)K:J4LMI�H[I�lwL\H0J4F3a4xiFkH0J4LMI�H5WEJ4ITS�UVLM`GL\H_hAI�X2JVUF3�0X_K�J4LMI�H_W�lSa4Ii}©J�DGF�}�I2Y2F3y�UEK�WVó_a�W�Je`_a4Ii`5I0W�FZY[v0�O�0J�a4I�J4ö�K�H_YXWèI�y)Y�Q-bZñQ�*RiTEK�H5YOyMK�J4F3aX_W4F3YõLMH/Y�L)W�DGF3a�Q-bZñ0ý9R0T|K�H_Y[�I�v;Fky^Q-b3ñiñQR0T�{>ôÂa�K�`GDGL)fkK�yoagK�}�L\ó5fkK�J4LMI�H_W|I�l�J4D_LMWEJ�a�K�H_W4yMK�J4LMI�HU|Fka�FÂFk�2`Gy\L)fkK�J4FZYAó_agW�JELMH(�`GLMa�J�F3WVFkJ�K�yq{�Q-bZñ�ñ��0T�K�H_YõyMK�J4F3a|LMHOð�F3K�a4y>Q-bZñ�ñi��v5T'{

Page 10: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

dbd � ÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê^=�A^euxZ]X c�´$Z]�9\`��dg÷b÷bÁ9¡±j���Þ�w{~�^P~�^ �D�9�]~���~�ux��uxZ}�«¡5�æ\=Z]zy\=Z]^`Z}a5ux~�a{�¯u`wyZ2�G�yay�4ux~�_ba» º ~�ux^`Z}���$�9^r�2|��9���yZP_��Æ�7|��9\`~��9¡y��Z�ª[Z º �9ay�¯uxwyZ}aµ��\=~�ux~�ay�µcÀÁ�� � j���^

>º !]\¬c%¼w½ º ��» º ��Ô º j�� cDÁ{�)�9j��wyZ}\`Z^\�~�^��7uxw{\`Z]Z��O�9\=�b�yX[Z]a5u��G�yay�4ux~�_ba¯^x��u`~�^=�Q�A~�ay�\�c�½>�:æ2�9_]j�!�» º c�½>�9_]j���wyZ}ayZ}|9Z]\õæm!¶» º ÀÞ�wy~�^P�9X[_b�ya5ux^©u`_v�]_ba{�]Z]z�ux�1�9��~�ä}~�ay��u`wyZ2~�a?u`Z]\e|bZ]a5u`~�_ba��9^���aæZ4�AuxZ}\`a1�9�$�G_b\`�}ZZ º u`w1��ur�9��uxZ]\=^�uxwyZo�G�{ay�}u`~�_baâ» º ¡UZ}ut��Z]Z}a¨ø º �9ay�¯~�u`^�z1�9\`Z}a5ux^]���P\x�9z{wy~��]�9�����bª��Z¤�]�9a�\=Z]zy\=Z]^=Z]a5u`Z º ��^���aM�9�{�yZ]�'z1�9\=Z]a5uHa{_��yZF_9�¥ø º ª��9ay��uxw{Z¤Z4­�Z}�}uS_9�^`�{��w«�9av~�a5uxZ]\e|bZ}a?u`~�_bav�]�9a¯¡¬Zo�9a1�9����ä}Z]�v¡5��^eu��9a{�1�9\`�¯�}_bay�y~�ux~�_9a1�9��~�äg��u`~�_ba?»uxwy��uS~�^]ª�¡5�'�]_bay�{~�ux~�_bay~�ay�v_b�y\Szy\`_9¡1�9¡y~���~�ut�¨_ba'uxwyZ[Z}|9Z]a5uHu`w1��uS|9��\`~¦��¡y��Z;Z º��u=u��9~�ay^�uxwyZP|��9���yZ?» º �

PA PA

GG

i i

iFiX Xi

è�~��b�y\`Z�Á�� � ÿ¾ê�Z]z{\`Z]^=Z]a5ux~�ay�ãZ��AuxZ]\=a1�9�P~�a?u`Z]\e|bZ]a5u`~�_ba]Z º ¡5�!�9aá�9�y�bX[Z]a5uxZ}�ayZ4ut��_b\=´�± = !̱�aÄÞbZ ºdc ø º ß?�Þ�wyZ�Z}­¬Z]�4u¨_��2�9a!��u`_bX[~��«~�a5uxZ]\e|bZ}a?u`~�_ba�êsë{cQø º ! =º j�~�^vZ]ay�}_��{Z]�!¡?��9�y�{~�ay��ux_©± ����~�ay´KZ º H c ø º cD^`Z}ZMè$~��b�y\=Z�Á{� � j4ªr��wyZ]\=ZeZ º ~�^¯��ayZ}�|��9\`~��9¡y��ZHu��9´?~�ay�µ|��9���yZ]^�~�aÜÞ_êsë�c =º j��S~��y��Z_ß?ª =º \x�9ay�9Z]^�_�|bZ]\�uxw{Z¤�y_9X��9~�aæ_9�ø º ª±�9a{�¤à=~��y��ZiáF\`Z]z{\`Z]^=Z]a5ux^ ay_[~�a5uxZ}\=|bZ}a5ux~�_9a/��Þ�w?�y^}ªUu`wyZ�ayZ}�ºz1�9\`Z}a5u©^=Z}ur_9�

ø º ~�a�uxwyZo�9�y�bX[Z]a5uxZ}��ayZ4ut��_b\=´�~�^��gf =º !��gf º a¬ÞbZ º ßs���9ay��~�u�~�^¥\`Z}�¦��u`Z]��u`_ø º ¡?��uxwyZ��]_9ay�y~�u`~�_bay�9��zy\=_b¡1�9¡{~���~�ut���c >º �2¼>½ =º j�!

hij ik ��c >º �G¼w½ º j�~��lZ º !P~��y��Z5�i ~��lZ º !�êsë{c =º j!�9ay� >ºnm! =º �d ~��lZ º !�êsë{c =º j!�9ay� >º ! =º ÀcDÁ{� fbj

Þ�wyZºZ4­�Z}�}uÜ_9�æu`wyZº~�a?u`Z]\e|bZ]a5u`~�_ba êsë�c =º jÍ~�^¼ux_�u`\x�9ay^e�G_b\`X u`wyZº_b\=~��b~�a1�9�zy\=_b¡1�9¡y~���~�ut���G�{ay�}u`~�_baÎ�¤c ���À8À8À�� >ä j×~�a5u`_ � a{Z}� zy\`_b¡y�9¡y~���~�ut� �G�yay�4ux~�_ba

Page 11: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ\É{ÎÍÓOÖ7É$Ê£ËAÊ�Ê£ÖFÉ�Ó4Ù�ÖÓOÖըǩËjËHÙ?Ê$ÓtÇoÖ*Õ,Ù�ÒPÊ¥Ñ�Ð dbdgÁ�¤c ���À8À8À�� >ä ��� =º j4ªy�b~�|bZ}a¨¡5�

�¤c ���0À�À8À�� >ä ��� =º j�!Ì� = c �:�À�À8À8� >ä �2Z º !Ìêsë�c =º j=j�� cDÁ{� ÷bj��wyZ}\`Z·� = ~�^�uxwyZ��y~�^eux\`~�¡y�{u`~�_ba¯^`zUZ]�}~�®1Z}�æ¡5�µu`wyZ2���y�bX[Z]a5uxZ}�«ayZ4ut��_b\=´�± = !±oaÌÞbZ ºpc ø º ß¼�9ay�ÚcDÁ{� fbj4ªP��~�u`wþ�9aþ�9\=¡y~�u`\x�9\e�!zy\`~�_b\¨�{~�^=u`\`~�¡y�{ux~�_ba!_baZ º �Ýstaã�bZ}ayZ]\`�9�Jª�¡5�ã���y�y~�a{�M�'w?�AzU_9uxw{Z}ux~��g�9��~�a?u`Z]\e|bZ]a5u`~�_baÍ��~�a{´qZ ºc ø ºux_�Z]�9��w¤a{_��yZ�~�a ±Fªb��Z �]�9aF�]_bay^eux\=�y�}u��9a[�9�y�bX[Z]a5u`Z]�[zy\`_b¡y�9¡y~���~�ut���G�yay�4ux~�_ba� = c ���0À�À8À�� >ä Û9Z����À8À�À8�9Z ä j¥u`w1��u��]_ba5u���~�ay^�~�a{�G_b\=X���u`~�_ba¤�9¡U_b�{u�\=~���wyZ}\�ut�Az¬Z}^�_9�~�a5u`Z]\=|9Z]a5ux~�_bay^}�,§¨�{��ux~�zy��Z¤~�a5uxZ}\=|9Z]a5ux~�_bay^2��_b�y����¡UZ�\=Z]zy\=Z]^`Z}a5uxZ]�Í¡5�,�]_9ay�y~��ux~�_bay~�ay�O� = _ba��P^=�y¡y^`Z4u�_��¬u`wyZrZ º cGux�9´?~�ay�o|��9���yZ]^£~�a2uxwyZ}~�\Y\`Z}^`zUZ]�4ux~�|9Zõêsë�c =º j�y_bX¤�9~�ay^�j�ª��9ay�áu`wyZMz{\`Z]~�a5uxZ]\e|bZ}a?u`~�_baþz{\`_b¡1��¡y~���~�ut�ã�G�{ay�}u`~�_baÌ� ��_b�y���º¡UZ|A~�Z4��Z}�ã�9^2u`wyZvzU_b^=u`Z]\`~�_b\2�y~�^=u`\`~�¡{�{ux~�_ba,~�a{�y�y�]Z}�Í¡5�M�]_bay�{~�ux~�_bay~�ay�¸Zg����wsZ º~�a¬� = _ba¯uxwyZP|��9���{Z¬à=~��y��Z9�dá²oayZ7�9�{|��9a5ux�9�bZ2_9�£u`wyZ7�9�y�9X¤Z}a?u`Z]�æayZ4ut��_9\`´¸\=Z]zy\=Z]^`Z}a5u���u`~�_baæ~�^ uxwy��uP~�u~�^F��zyzy��~��g�9¡{��Z¤u`_¼>9K13���w1�9a{�bZ¯~�a�uxwyZ��G�{ay�}u`~�_ba1����\`Z}�¦��u`~�_bay^=wy~�zÕ» º �9ay�¾ay_�uX[Z]\`Z}���vu`_�uxwyZ�\`Z}zy�¦�9�}Z]X[Z]a5uo_���» º ¡5�«�¤�}_bay^=ux�9a5ug�osâuo�9��^=_��y~�^`zy���g��^ �]��Zg�9\=���uxw{Z \`�9X¤~�®1�g�Lux~�_9ay^Y_9�/^=z¬_9a?ux�9ayZ}_b�y^���wy�9ay�bZ}^�~�a » º ª?�yayX[Z]�y~���uxZ}�¤¡5�[Z4�AuxZ}\`a1����]_9a?u`\`_b�À�©è$~��b�y\=ZHÁ�� � zy\`Z}�y~��4ux^]ªU�G_b\©Z��{�9X¤z{��Z9ª±uxwy��u©_9ay���µ�yZ}^`�]Z}ay�1�9a5u`^P_9�Yø º��_b�y���µ¡UZ�Z4­�Z}�}uxZ}�«¡5�¯��w1�9a{�bZ]^ ~�a�» º �9ay�µwyZ]a{�]Z�u`wyZSX��9\=�b~�ay�9��zy\=_b¡1�9¡{~���~�ut��¤c4/5j���~����¥\`Z]X¤�9~�a��ya1�9��uxZ]\=Z]�,�G_b\7Z4|bZ}\=��^=Z}u�ú�_9� ay_bay�yZ}^`�}Z]ay�1��a?u`^¤_�� ø º ��°~�´�Z}��~�^=Z9ªYè$~��b�y\=Z[Á{� � �y~��4u���u`Z]^�uxw1�Lu�u`wyZ��}_bay�y~�ux~�_9a1�9�Yzy\=_b¡1�9¡y~���~�ut� �¤c���� >º j\`Z}X���~�ay^r~�a5|��9\`~��9a5u�ux_���w1��ay�bZ]^©~�a » º �G_b\ �9a5�¨^`Z}uoù�_9�£�{Z]^`�}Z]ay�y�9a5ux^P_9�Yø º ªzy\=_;|A~��yZ]�Mø º ê9�t^`Z}z1�9\`��uxZ}^tZ º �G\`_9X ù¤�AuPZ}|A~�aȳ _A_;|9Z]\�ced�÷b÷9i{ª�dg÷b÷b÷5jo�y^=Z]�^`�{��w�~�a?|��9\=~¦�9a5uo�GZ]��ux�{\`Z]^ou`_¯�yZ}u`Z]\`X[~�a{Z2uxwyZ7�y~�\`Z}�}ux~�_baæ_9�¥�g���y^x�9�Y~�a��1�yZ}ay�]Z}^�9X[_bay�oZ}�]_ba{_bX¤~���|��9\=~¦�9¡y��Z]^�cDZ9�ý�y��ªbZ]X[zy��_;�AX[Z]a5u£�9a{�7X¤_ba{Z}��^`�yzyz{���{j�¡5�2_b¡��^`Z}\=|A~�a{�¯u`wyZ7��w1�9a{�bZ]^�~�a{�y�y�]Z}��¡?�¸^`�y�y�{Z]a'X¤_A�y~�®y�g��u`~�_bay^ ~�a¸u`wyZ7zy\`_A�]Z}^`^=Z]^uxwy��u2�b_�|9Z]\`a,u`wyZ]^=Z�|��9\`~��9¡y��Z}^�cGZ9�ý�y��ª�u��L�M\`Z4�G_b\`Xµª����9¡U_b\��{~�^`z{�{uxZgj4�«sta{�yZ]Z}�/ª��wyZ}ayZ}|9Z]\o��Z2_b¡{ux�9~�a¨\`Z]��~¦�9¡{��Z�~�a{�G_9\`X¤��ux~�_ba'cDZ��Â�y��ª¬�G\=_bX wy~�^eux_b\=~��]�9�°_b\©~�ay^=u`~��ux��ux~�_9a1�9��´?ay_�����Z]�y�bZgj©uxw1��uS��a���¡y\`�yz�u���_��]�9����wy�9ay�bZFw1�9^Pux�9´9Z}a�zy���9�]Z7~�a��^`zUZ]�}~�®1� X¤Z}��w1�9ay~�^`X » º uxw1�Lu��]_9ay^=u`\x�9~�ay^����b~�|9Z]a��D�9X[~����¯cQø º ���gf º j£_9�°|��9\=~���9¡y��Z]^}ª?��Z©�]�9a��{^`Z u`wyZr_b¡y^`Z}\=|9Z]����w1��ay�bZ]^�~�aFuxwyZrX¤�9\`�b~�a1�9�1��ay���}_bay�y~�ux~�_9a1�9�zy\=_b¡1�9¡y~���~�ux~�Z}^�^`�y\=\`_b�ya{�y~�ay� uxwy_b^=Z�|��9\`~��9¡y��Z}^�u`_o�yZ4uxZ]\=X¤~�ayZ���w{Z}uxw{Z]\Æø º ~�^$~�a���yZ}Z]��uxw{Z©��wy~����æcG_b\¥�yZ]zUZ]ay�{Z]a5u�|��9\`~��9¡y��ZgjÆ_9�/uxwy��u��D�9X[~����bªbu`wA�{^��yZ}u`Z]\=X¤~�ay~�ay�uxw{Zv�y~�\=Z]�}u`~�_ba�_�� �]�9�y^`�9��~�a��1�yZ]a{�]Z]^F~�a,uxwyZv�y_9X��9~�a/��Þ�wyZ¯^=ux��ux~�^=ux~��g�9�¥�GZg�L�ux�{\`Z]^�uxwy��u�\`Z}X��9~�a�~�a5|��9\`~��9a5uP�yay�yZ}\�^=�y��wM��wy�9ay�bZ}^]ªÆ��^���Z]���£�9^PuxwyZF�]�9�y^x����9^=^`�yX[z{ux~�_bay^Æ�{ay�yZ]\=���A~�a{�©uxwy~�^Æ~�a5|��9\=~¦�9ay�}Z9ª5��\`Z��y~�^`zy���g�bZ]�7~�aHuxw{Z��9�y�bX[Z]a5uxZ}�ayZ4ut��_b\=´�± = �

Page 12: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

dbdgÃ1ÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê�m�����g� v ¥nw¹�Xxj®>°k­?y ®� [§{z}| §�~|®¤¥�ª�«s­�®�§���¯�§.­�®�°'¥�­[¡ê�Z]�5�9\=�y��Z}^`^�_9�¥��wyZ}u`wyZ]\���Z[\=Z]zy\=Z]^=Z]a5u2~�a?u`Z]\e|bZ]a5u`~�_bay^S�9^S�vX[_��y~�®1�g�Lux~�_9a¸_9��9a+Z���~�^=u`~�ay��X¤_A�yZ}�Hc�²oZ}®1a{~�ux~�_baÍÁ{� � ��d;j7_b\��9^��'�]_bay�{~�ux~�_ba1�9��~�ä]��ux~�_ba�~�aã�9a�9�y�9X¤Z}a?u`Z]�[X¤_A�yZ}��cDZ]¢?�1�Lux~�_9a¸cDÁ{� ÷bj`j4ªbu`wyZr\`Z]^=�y��u¥~�^£����Z]�����t�{Z}®1ayZ}�¤u`\x�9ay^e�G_b\e�X¤��ux~�_ba2¡UZ}ut��Z]Z]a¤uxwyZ�zy\=Z]~�a5u`Z]\=|9Z]a5ux~�_ba¤�9ay�Fz¬_9^=ux~�a5uxZ]\e|bZ}a?u`~�_ba[�y~�^=ux\=~�¡y��ux~�_9ay^]�staãu`wyZ«�]�9^`Z¨_9���9aÝ�Lux_bX[~��¯~�a5u`Z]\=|9Z]a5ux~�_baÃêsë�cGø º ! =º j�ª�uxwy~�^¤u`\x�9ay^e�G_b\`X¤�L�ux~�_ba«�]�9a¸¡UZHZ��{z{\`Z]^=^`Z}��~�a¸��^=~�X[zy��ZFBDß4çAK�@`>9Bt0`T;Ø N}>A@4BO6Lß�ICB�>9BDIJ6LK��G_b\`XH�y�¦�7uxw1�Lu�G_b����_���^�~�X[X[Z]�y~���uxZ}���F�G\`_bX�cÀÁ{� � j��9a{��²oZ4®1ay~�u`~�_ba¯Á�� � ��dbÿ��

��c ���À8À�À8� >ä ��� >=º j"!���� ��� º ��c �� ¼w½ j�~�� >º ! =º �i ~�� >ºDm! =º À cÀÁ{��d�i5jWY¢?�1��ux~�_baãcÀÁ{��d�i5j©\=Zi�1Z}�}ux^SuxwyZ[\`Z}X¤_�|��9�Æ_���uxw{Z[u`Z]\`X ��c >º �s¼w½ º j©�G\=_bX�uxw{Zzy\=_��y�{�}uF_9�2cÀÁ{� ìbj�ª¥^=~�ay�}Z·¼w½ º ay_���_bay�9Z]\7~�a��1�yZ]a{�]Zvø º ��è1_b\2Z��{�9X¤z{��Z9ª£uxw{Z~�a5u`Z]\=|9Z]a5ux~�_ba êsë�cGø ! = j/��~����9u`\x�9a{^=�G_b\=X×uxwyZ¥zy\=Z4�t~�a5uxZ]\e|bZ}a?u`~�_ba��y~�^eux\=~�¡y�{u`~�_ba�b~�|bZ]aµ~�a�cDÁ{� hbj�~�a5ux_2u`wyZ�zy\`_A�y�y�4u

��c�/gü�:/5����æG��/�2�:/������� >= j"! ��c�/gü4j3�¤c4/5�2� /gü�j3�¤c4æ�� /gü�j6 �¤c4/��� = �:/5�`jk��c�/;�</�0�:æ4j3�¤c���� = �:/�2�:/�j�À

�P\x�9zyw{~��g�������9ª�u`wyZ�\=Z]X[_�|9����_���uxw{ZSu`Z]\`X �¤c >º � ¼w½ º j�~�^ Z}¢?�y~�|��9��Z]a5u�ux_[\`Z]X[_�|5�~�a{��u`wyZ���~�ay´?^Y¡¬Z4ut��Z}Z]a �gf º �9a{�[ø º ��wy~���Z�´9Z}Z]zy~�ay�Su`wyZ�\`Z}^=u�_9�¬u`wyZ�ayZ}ut��_b\=´~�a5ux�9�}u]� · ��Z]�9\`���bª5uxwyZrux\`�9ay^=�G_9\`X¤��ux~�_baF�yZ}®1a{Z]�v~�a«cÀÁ���d�ibj£^`��ux~�^=®1Z}^�uxw{Z©�]_9ay�y~��ux~�_ba¯_9�²oZ}®1ay~�ux~�_ba¸db�ÂÁ{��dH�9^���Z]�����9^�uxwyZ�zy\=_bzUZ]\=u`~�Z}^�_9��cOdb�ÂÁbf5jV�¬cedb�ÂÁb÷5j��

§¨�y��ux~�z{���A~�ay� �9ay�H�y~�|A~��{~�ay�HcÀÁ{��d�i5j°¡5� �¤c =º � ¼w½ º j4ª�u`wyZ�\=Z]�¦�Lux~�_9ay^`wy~�zSu`_©uxw{Zzy\=Z]~�a5u`Z]\=|9Z]a5ux~�_baµ�y~�^eux\`~�¡y�{u`~�_bav¡UZ]�}_bX¤Z}^�X¤_b\=Zoux\`�9ay^`zy�9\`Z}a?u]ÿ

�¤c �:�ÀÀÀi� >ä �4� >=º jÎ! hj k �¤c ���0ÀÀÀ0� >ä j�¤c =º �2¼w½ º j ~�� >º ! =º �i ~�� >ºDm! =º À

cÀÁ{��dbd;j

sâ�£��ZH\`Z]�b�9\`����³=_9~�a5uo�y~�^=u`\`~�¡{�{ux~�_ba¸�9^o�9aæ�9^=^`~��9ayX¤Z}a5uP_9�£X¤�9^`^©u`_v���}_b����Z]�4�ux~�_baæ_9���9¡y^eux\`�9�}u�zU_b~�a5u`^[c ���À0ÀÀ0� >ä j4ª°Z]�9��w'\`Z]z{\`Z]^=Z]a5ux~�ay�¨��z¬_b^=^`~�¡y��Z7^eu���u`Z�F� �0X_K�J4LMI�HAQ8�_{\bFRiTof3K�H�K�yMW4I�v;F�I�vGJ�K�LMHGFZYÂlSa4Ii}âJ�DGF��ÂúÙf'Ii}�`GX2J�K�J4LMI�H�lSIia4}�XGyMK�I�lGüVI�v_L\H_WQ-bZñQ�*�G~�`s{zb3�0÷��Gc�W�F3F �F3f'J4LMI�H �G{ �G{ �0T[K�H_Y�J4D_F�}mK�HGLM`GXGy)K:J�L\IiHâJ�DGFkIia4F3} I�lm�2`GL\a4J4FZW(FkJK�yq{wQ-bZñ�ñi�iT�Q8K�f3f'Iia�Y2LMHGhjJ4IõJ4D_LMW�W4I�XGagf'Fi~�W�K�L)YOlSI�a�}zX_yMKõUEK�W/S�LMH_Y2F3`5F3H_Y2FkH0J�y\�?fkI�H9�-FZf�J�XGa4FZYv��Y�LMFkHv;Fka�h(L\H�K?W�F3}�LMH_K�azLMH�b3ñiñGb�U0T�{[�.YGYGLdJ�L\IiH_K�y^`Ga4Ii`5F3a�J�L\FZW�I�l|J�DGFAJ�a�K�H_W�lSI�a�}�K�J4LMI�HY2Fkó_HGF3Y�L\H¬Q��G{Mb�R0T�K�H5Y Q8�G{Mb�b�T�K�a�FAhiL\xiFkH·LMHÑôÂI�y)YGW4ök}�L)YJ�K�H_Y·ð�F3K�a4yÂQ-b3ñiñi÷iT�K�H_Y�ð�FZK�a�yQ-bZñ�ñi��v5T'{

Page 13: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ\É{ÎÍÓOÖ7É$Ê£ËAÊ�Ê£ÖFÉ�Ó4Ù�ÖÓOÖըǩËjËHÙ?Ê$ÓtÇoÖ*Õ,Ù�ÒPÊ¥Ñ�Ð dbd�ì_9�$uxwyZ���_9\`����ª±u`wyZ]a¨uxwyZ�ux\`�9ay^=�G_9\`X¤��ux~�_bav�yZ]^=�]\=~�¡UZ]�¸~�a,cDÁ{��d9d;j�\`Z}|9Zg�9��^ ^=_bX[Z~�a5u`Z]\`Z}^=u`~�ay�Pzy\=_bzUZ]\=u`~�Z}^Æ_9�1uxw{Z���w1�9ay�bZ�~�a2X¤�9^`^Æ�{~�^=u`\`~�¡y�{ux~�_ba�u`w1��uÆux�9´9Z�z{�¦�9�}Z�9^£�P\`Z}^`�y��u£_9���9aF~�a5uxZ}\=|bZ}a5ux~�_9aÑêsë�cGø º ! =º j�c4��_b���y^`ä}X¤~��{uY��ay�Ý´$Zg��\`�°d�÷b÷ � j4�W£�9��w�zU_b~�a5u�c ���À0ÀÀ0� >ä jP~�^S^`Z]Z}a,ux_¸~�ay�]\=Zg�9^=Z�~�ux^SX��9^=^�¡5�M�µ�D�9�4ux_b\SZ]¢?�1���ux_�uxw{Z«~�a5|9Z]\`^=Z«_9� uxwyZµ�]_9ay�y~�u`~�_bay�9��zy\`_9¡1�9¡y~���~�ut�È�¤c =º � ¼w½ º j7�]_b\=\`Z}^`zU_bay�y~�ay�ux_Íuxw1��u«z¬_9~�a5ug� ´$_b~�a?u`^«�G_b\µ��wy~���wºu`wy~�^¨�]_bay�{~�ux~�_ba1�9��zy\=_b¡1�9¡{~���~�ut�Ý~�^¨��_����_b�y���Í¡¬_A_b^eu¤u`wyZ]~�\[X¤�9^=^[|��9���yZ¨^=�y¡y^eu��9a5ux~��9�����bª£��wy~���Zµu`wy_b^=Z¨zU_b^`^=Z]^=^`~�a{���¼w½ º |��9���yZFuxw1��uH�9a5ux~��]~�z1��uxZ}^H�µa1��u`�y\x�9�©cDa{_bay~�a5u`Z]\=|9Z]a5ux~�_ba1�9�¦j�\=Zg�9��~�ä]��ux~�_baæ_9� =º cG~J�ýZ9��ªm�¤c =º � ¼w½ º j�� d;j2��~�����´9Z}Z]z+uxw{Z]~�\FX¤�9^`^¤�ya1�9��uxZ}\`Z]���Ýsta+^=ux�9ay�1�9\=�ã��g�bZ}^��}_bay�y~�ux~�_9a1�9��~�äg��u`~�_ba�ªÆZg�9��wÍZ4���]���y�yZ}�+z¬_9~�a5u«c >º�m! =º jHux\`�9ay^=�GZ}\`^[~�u`^X¤�9^`^2u`_¸uxwyZ¯Z]a5u`~�\`Z�^`Z}u[_9�rzy\=Z]^=Z]\=|9Z]�+zU_b~�a?u`^Huxw{\`_b�y�9wÍ�¸\=Z]ay_9\`X¤�9��~�äg��u`~�_ba�]_9ay^=ux�9a5ug�Õ³r_;��Z}|9Z]\}ª�cÀÁ{��dbd;jS�yZ]^=�]\`~�¡¬Z}^[���{~�­¬Z]\`Z}a5u7ux\x��ay^=�G_b\=X��Lux~�_9a/ÿ¤Z]�9��wZ4���}���y�yZ}�+zU_b~�a5u¸c >º}m! =º j2ux\`�9ay^e�GZ]\`^�~�ux^FX¤�9^`^[ux_M��^`Z}��Z}�}u¤^`Z}u�_9�PzU_b~�a5u`^uxwy��u�^`w1��\`ZPuxwyZP^`�9X[Zo|9�����yZP_9�w¼w½ º �YÞ�w{~�^��g��aµ¡UZ�^`Z}Z]a¯�G\`_bX uxwyZP�}_bay^=ux�9ay�4�_9��¡U_9uxw�uxwyZ[ux_9ux�9��X¤�9^=^2�9^=^`~��9ayZ]�,u`_«Zg����w¼^eux\`��ux�yX ¼w½ º ��ay�Muxw{Z�\`Z}�¦��u`~�|bZX¤�9^`^=Z]^�_9�$z¬_b~�a5ux^���~�u`wy~�avZ]�9��w«^=�y��w«^eux\x�Lux�yXµÿ

��c%¼w½ º �<êsë�c >=º j`jÎ! �¤cS¼w½ º j4Û�¤c�� º �4¼w½ º �<êsë�c =º j`j�¤c�� =º �4¼w½ º �<êsë�c =º j`j ! �¤c�� º �4¼w½ º j�¤c�� =º �4¼w½ º j À³ Z}\`Z}� º �yZ]a{_9uxZ}^Fu`wyZ¯^`Z4u¤_��o�9����|��9\=~¦�9¡y��Z]^2Z����]���{�y~�ay�ÕÞ_�gf º a�ø º ß?��Þ�wy~�^^`Z}��Z}�}uP^=Z}u�_9�£X¤�9^`^O�t\=Z]�]Z}~�|A~�a{��z¬_9~�a5ux^ �g�9aæ¡UZ2\`Z}�5�9\=�yZ]���9^ à=�]��_b^`Z}^=u�á¤ux_�uxw{ZzU_b~�a5u�Z4���]���y�yZ]�¾¡5�'|�~�\=u`�yZ�_9��^=w1�9\=~�ay�«u`wyZ�^`�9X[Z�wy~�^eux_b\e�bªY��^H^`�{X¤X¤�9\`~�ä]Z}�¡5� ¼w½ º cD^=Z]ZH¿�Z]�4ux~�_9ay^�Ãy��d9� ÁF�9ay�����ýÃy�ÂÁ5j4�½©ay_�uxwyZ}\[~�a5uxZ]\=Z]^eux~�a{���G_9\`X¹_��7cDÁ{��d9d;jH_b¡{ux�9~�ay^2��w{Z]a¾��Zµ~�a5uxZ}\`zy\=Z}uFuxw{Z�y~�|�~�^`~�_bav¡5����c =º � ¼w½ º j��9^��]_ba{�y~�u`~�_ba1����~�ä]��ux~�_ba[_ba =º ��ay�ݼ>½ º ÿ�¤c ���0ÀÀÀ0� >ä ��� >=º j�!�� �¤c �:�ÀÀÀi� >ä � =º ��¼>½ º j3�¤cS¼w½ º j ~�� >º ! =º �i ~�� >ºnm! =º À cÀÁ{��d � jÞ�wy~�^©�G_b\`XH�y�¦��¡UZ]�}_bX[Z]^©�D�9X[~���~¦�9\���w{Z]a��y^`Z}�'u`_v�]_bX[zy�{u`ZHuxw{Z2Z}­¬Z]�4u�_9���9a~�a5u`Z]\=|9Z]a5ux~�_ba�êsë{cQø º ! =º j¥_9a¯�H^`Z4u�_9�°|��9\`~��9¡y��Z}^�ù*�y~�^Q³=_b~�a5u�_���cGø º a¸�^f º j4�¿��yX[X[~�ay�McÀÁ{��d � jP_�|bZ}\F�9���Y|��9\`~��9¡y��Z]^�Z4���]Z}z{u[ù{a�ø º �A~�Z]���{^�uxw{Z��G_b����_���~�ay�uxw{Z]_b\=Z]Xµ���� ����b 2� ����������!F���2� &�(*�I�e 2���:¡V���¢�A���I 213�£#�%¤&)(� 2(b4¥�04BV�gf º TA04K�6LBt0rBG.10�:�04B�6ON©T9IQßx0�@4B¥@`>9ç5:�0�:�6ON�àL>9ß�ID>58]C�0�ø º 8£>9KUTHC�04B°ù�840P>9K13

Page 14: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

dbd�hyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê:�04B©6ON2àL>9ß�ID>58]C�0�:7T9I�:�7g6LIQK1B©6ONèÞ�ø º aÄ�gf º ß�V'-/.10�0�¦o0�@4B©6ON�BQ.10FIQK1BO04ß�à904K1BDIJ6LKêsë{cQø º ! =º j26LK¯ù&I�:õ�9IQà904K�8]3

��c-�.��� >=º j�!�# §$¨ª©��¤c���� >=º ��¼w½ º jk��c%¼w½ º j�� cÀÁ{��d�Á5j6�.104ßx01�¤c���� =º ��¼w½ º j�>9KUTè��c%¼w½ º j�ß�0>=Ußx0�:�04K1B�=Ußx04IQK1BO04ß�à904K1BDIJ6LKT=Ußx698�>58]IQCEIQBÀIJ0�:]VWY¢?�1��ux~�_ba�cÀÁ{��d�Á5j��]�9����^��G_b\¯�]_bay�{~�ux~�_bay~�ay�Ü��c-�.� =º j�_baÜuxw{Z�z1��\`Z]a5u`^¨_9��ø º�9ay�Üuxw{Z]aº�g|9Z]\x���b~�ay�¾uxw{Z'\=Z]^=�y��u]ªo��Z}~��bw5uxZ}�ס5�Ýuxw{Z'z{\`~�_9\vzy\`_b¡y�9¡y~���~�ut�Í_9��gf º ! ¼w½ º �PÞ�w{Z7_bz¬Z}\x��u`~�_ba¨�yZ}®yayZ]��¡5�«u`wy~�^ �]_bay�{~�ux~�_bay~�ay����ay���g|bZ]\`�9�b~�ay�~�^�´?ay_���a«�9^ à`�9�L³=�y^eux~�ay�7�G_b\A�^f º �dáµ¥�9\`~���ux~�_bay^Y_9�¬uxwy~�^¥�9�L³=�y^euxX[Z]a5u�wy��|9Z©¡UZ]Z}a��9�{|��9a{�]Z]�¤¡5�FX��9a5�Fzyw{~���_9^`_��zywyZ}\`^��9^�z{\`_b¡1��¡y~���~�^=u`~����{Z}®1ay~�ux~�_bay^£_9�/�]�9�y^`�9��~�ut�F�9ay�¤�g�9�y^`�9�±Z}­¬Z]�4u�cD^=Z]Zo¿�Z]���ux~�_ba·�A� ìbj����P_�_A�vcOd�÷bh{d�j4ª��G_9\$Z��y��X¤zy��Z9ªb�]�9����^��G_9\��]_9ay�y~�u`~�_ba{~�ay�r_baÄàeu`wyZ�^eu���u`Z_9�/u`wyZo�ya{~�|bZ}\`^=Z¥³=�y^=u�¡¬Z4�G_b\`Ziá�u`wyZo_A�]�]�{\`\`Z}ay�]Z©_9�/uxw{Z©�g���y^`Z���¿A�yzyzUZ]^�cOd�÷o�9i5j�g������^��G_b\2�}_bay�y~�ux~�_bay~�a{�¨_ba,u`wyZ�Z}a?u`~�\=Z�z1�9^eugª£�yz,u`_«uxwyZ�_��}�]�y\=\`Z]a{�]Zv_9��uxw{Z�g���y^`Z���¿A´?�A\=X¤^oced�÷bf9i{ª�z��UdgÁbÁ5jY�g�9����^£�G_b\��}_bay�y~�ux~�_bay~�a{�S_9aÜà=X¤�L��~�X¤�9�����H^=z¬Z���]~�®1�P^`zUZ]�}~�®1�]��ux~�_bay^�_9��u`wyZP�D�9�}u`_b\`^�_b��ux^`~��yZP_9��_b�y\�~�a��1�yZ}ay�]ZS��u�uxwyZPux~�X¤Z _9�uxw{ZH�yZ}�]~�^=~�_baµ��wy~���w¸�9\`ZS�g�9�{^x�9�����¯\`Z}��Z4|9��a?urux_[uxwyZ�_b�{u`�]_bX[Z�_��Y_b�y\ �9�}u`~�_ba{^�����dáy��Þ�wyZF�9~�X&_9�£�}_bay�y~�ux~�_bay~�a{��~�a¨uxwyZ}^`ZFzy\`_9z¬_b^`�9��^ ~�^}ª/_9�¥�]_b�{\`^`Z�ª/ux_vZ}��~�X7�~�ay��uxZ�^=zy�y\=~�_b�y^7�}_b\`\=Z]����ux~�_bay^2¡UZ}ut��Z]Z}aÍuxw{Zv�g�9�y^=Z�cD~�a�_9�y\7�g�9^=Z9ª£ø º ! =º j�9ay�+u`wyZ«Z4­�Z}�}u�cDù !@�{j4Û©�}��Z]�9\`���bª�uxw{Z«^`Z4u¯_9��z1�9\=Z]a5ux^¸�^f º �g��aÜ�9�]�}_bXF�zy��~�^`w¸u`wy~�^���~�X&��~�uxwæ�b\=Zg��u�Z}�]_ba{_bXH�b�7sta�uxw{Z[^eux\=�y�}u`�y\x�9�£���]�]_9�ya5uSu`w1��uP��Zzy�y\=^`�yZ�~�a�u`wy~�^S¡U_�_b´±ª$�g�9�y^`�9�¥Z}­¬Z]�}u`^7�9\=Z��yZ4®1ayZ]��~�a,�µ\x�9�y~��g�9�����æ�{~�­¬Z]\`Z}a5u�����9�YÞ�wyZ��]_ba{�y~�u`~�_bay~�ay�©_9z¬Z}\x��u`_b\Y~�^Æa{_9u£~�a5ux\=_��{�y�]Z}�[~�a5ux_[cÀÁ���d�ÁbjÆ�9^£�P\`Z]X[Z4��y~��9�jà`�9�L³=�y^euxX[Z]a5u�á��9~�X[Z]�«��u©Z}\x�9�{~��g�Lux~�a{�¤^=zy�y\`~�_b�y^o�}_b\`\=Z]����ux~�_bay^]�rêr��uxw{Z]\]ª~�uSZ}X¤Z}\`�bZ}^��G_b\=X��9�����µ�G\`_bX�u`wyZ[�yZ]Z}z¬Z}\2zy\`~�ay�]~�zy��Z7\=Z]zy\=Z]^=Z]a5uxZ}�¼~�a+cDÁ{��d�i5jâ»uxwy��uY_9�Uzy\=Z]^`Z}\=|A~�a{�������?uxwyZ�~�a?|��9\=~¦�9a5u$~�a{�G_9\`X¤��ux~�_baSu`w1��uÆu`wyZ�zy\=Z]~�a5u`Z]\=|9Z]a5ux~�_ba�y~�^=ux\=~�¡y��ux~�_9av�g�9a¯zy\=_;|A~��yZ9�

Þ�wyZoux\`�9ay^e�G_b\`X¤��ux~�_ba[_9��cÀÁ{��d�i5j£�g�9avZ]�9^`~����[¡UZoZ4�AuxZ}ay�yZ}�¯u`_7X¤_9\`Z©Z}�¦�9¡U_��\x�LuxZ¥~�a5uxZ}\=|bZ}a5ux~�_9ay^�~�a���wy~���wH^`Z4|bZ}\x�9�A|9��\`~¦��¡y��Z}^$��\`Z�X¤�9ay~�zy�y����uxZ}�S^=~�X2�{��u���ayZ4�_b�y^=���9��è1_b\°Z��y��X¤zy��Z9ª;~��A��Z��}_bay^`~��yZ]\°u`wyZ£�]_bX[z¬_9�yay��~�a5uxZ}\=|9Z]a5ux~�_ba êsë�c��È!]�;j��wyZ}\`Zg�,~�^���^=�y¡y^=Z}u�_9�/|��9\=~¦�9¡{��Z]^}ª5uxwyZ}a'cDZ]��w{_b~�ay��cedb�ÂÁo�bj=jY��Zo^=wy_b�y�����yZ]��Z}u`Z�G\`_9X u`wyZ�zy\=_��{�y�}uH_9��cÀÁ{� ìbj��9���Y�D�9�4ux_b\=^·�¤c >º � ¼w½ º j��]_b\=\`Z]^=z¬_9ay�y~�a{�«ux_«|��9\=~���9¡y��Z]^�~�ap�Í�9ay�¯_b¡{ux�9~�avuxw{ZSX[_b\=ZP�bZ]ayZ}\x�9�/u`\`�ya{�g��u`Z]�¯�D�9�}u`_b\`~�äg��u`~�_ba�¤c ���À0ÀÀ0� >ä �-��;j�! hij ik åº�« Ï © �¬�­ �¤c >º � ¼w½ º j��G_9\ ���ÀÀÀi� >ä �]_9ay^`~�^=uxZ}a5u ��~�uxw}���

i _�uxwyZ}\=��~�^=Z;À cÀÁ{��dgÃ?j

Page 15: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ\É{ÎÍÓOÖ7É$Ê£ËAÊ�Ê£ÖFÉ�Ó4Ù�ÖÓOÖըǩËjËHÙ?Ê$ÓtÇoÖ*Õ,Ù�ÒPÊ¥Ñ�Ð dbd2��°~�´9Z4��~�^=Z9ª���Z¥ayZ]Z}�2ay_9u���~�X¤~�u/_b�{\`^`Z}��|9Z]^�ux_r^`~�X[zy��ZY~�a5uxZ}\=|bZ}a5ux~�_9ay^�u`w1��u�^`Z4u

|��9\`~��9¡y��Z]^$u`_o�]_9ay^=ux�9a5ux^]�Ysta{^=uxZ]�9�/ªb��Z�X��g�S�]_9ay^`~��yZ]\Y�oX[_b\=Z��bZ}ayZ]\`�9�{X[_��{~�®{��g�Lux~�_9a¤_9�¬uxwyZ �g�9�{^x�9�±X[_��yZ}�y��wyZ]\=Z]¡5�¤^`_bX[Z X[Z]��wy�9ay~�^=X¤^��9\=Z�ßx0>=UC�>A@�0`T9�Æè1_b\Z4�{�9X[zy��Z9ª±~�����Z�\`Z}zy�¦�9�}ZPuxwyZSX¤Z}��w1�9ay~�^`X�u`w1��u �yZ}u`Z]\`X[~�a{Z]^�uxwyZ�|��9���yZ�_9�$ø º¡5�¯�9ay_�uxwyZ}\rZ}¢?�1��ux~�_ba/ªy_9ayZPuxw1��ur~�a?|9_b��|9Z]^�zUZ]\`wy�9zy^r�7ayZ}�þ^=Z}uj�gf?®º _9�$|��9\=~���9¡y��Z]^}ª9uxwyZ}a7uxwyZ�\=Z]^=�y��ux�9a5uY�y~�^eux\=~�¡y�{u`~�_baS��_9�y���2_9¡{u��9~�a2¡5�2\=Z]zy���9�]~�ay� u`wyZ��D�9�4�ux_9\e�¤c >º � ¼w½ º j/��~�uxw�uxwyZ¥�]_9ay�y~�u`~�_bay�9�9zy\=_b¡1�9¡{~���~�ut�O� ® c >º � ¼w½ ®º j°~�a{�y�y�]Z}��¡5��uxw{ZayZ4�þZ]¢?�1��u`~�_ba���Þ�wyZ�X¤_A�y~�®1Z]�H³=_b~�a?u��{~�^=u`\`~�¡y�{ux~�_ba���_b�y���¯uxw{Z]a«¡UZS�b~�|9Z]aµ¡?�� ® c ���À0ÀÀi� >ä j�!Ì�¤c ���0ÀÀÀ0� >ä jk� ® c >º � ¼w½ ®º jF¯;�¤c >º � ¼w½ º j�À���s,/°�%'�e±)+� �²X³^�b��1´ µ(�(�#�������I��+Þ�_P~������y^=u`\x��u`Z¥u`wyZ]^=Z�_bz¬Z}\x��u`~�_ba{^]ªb��Z}uÆ�{^Y�]_bay^=~��yZ}\£�9a7Z4�{�9X[zy��Z�~�a5|b_9��|A~�ay�©zy\`_���]Z}^`^£�]_9a?u`\`_b�ÀÛ?�9a1�9��_b�b_b�{^Y�9zyzy��~��]��ux~�_bay^�~�a2uxw{Z��9\`Z]�9^Y_9�¬wyZg�9��uxw7X¤�9a1�9�9Z]X[Z]a5ugªZ]�}_bay_bX[~�� z¬_9��~��4�FX���´A~�ay�yª?zy\=_��y�{�}u�X¤�9\`´�Z}u`~�ay�yªA_b\�\`_b¡U_9u�X[_9u`~�_ba[zy�¦��ayay~�a{�^`w{_b�y�����G_b����_��~�a��¯^=ux\`�9~��9w?u=�G_b\=���9\=�����g�b� �°Z}u�uxwyZ7|��9\=~¦�9¡y��ZFú·¶7^=u���ay�'�G_9\uxw{Zµ^eu���u`Zv_9�©�¸zy\`_A�y�y�4ux~�_9a¼z{\`_A�]Z]^=^[��u7ux~�X¤Z:¸¹¶9ª��9ay�¾��Z4u2ø:¶v^=u���ay�¼�G_9\7�^`Z4uF_9��|��9\`~��9¡y��Z]^vcD��u2u`~�X[ZT¸¹¶�j�uxw1��u2~�^H�{^`Z]��u`_æ�]_ba5ux\=_b��u`w1��u7zy\=_��}Z]^=^µcD^`Z}Zè�~��b�y\`Z�Á{� Ábj4��è1_b\7Z��y��X¤zy��Z9ª�ú·¶v�]_b�y����^=ux�9ay�¾�G_b\7^`�y��wÍX¤Z]�9^`�{\`Z]X[Z]a5u`^¤��^

Y OUTCOME

XnXk1X -1kX

Zk ZnZ1 k-1Z STATES

CONTROLS

è�~��b�y\`Z2Á{� Á�ÿO²©�Aa1�9X[~��2�]�9�y^x���Y�y~¦�9�9\x�9X�~������y^eux\x�Lux~�a{�¤ut�Azy~��]�9�Æ�yZ}z¬Z}ay�yZ}ay�]~�Z]^�9X[_bay�SuxwyZo�}_ba5ux\=_b�U|��9\=~¦�9¡{��Z]^£ø ���ÀÀÀi�`ø ä ª�uxw{Zo^=ux��uxZr|9��\`~¦��¡y��Z}^�ú����ÀÀÀ���ú ä ª�9ay�¯uxw{ZS_9�{ux�}_bX¤Zo|��9\=~¦�9¡{��ZPù _9�Æ�7^`Z]¢?�yZ}a5ux~¦����z{\`_A�]Z]^=^]�uxZ}X¤zUZ]\`��ux�{\`Z���ay�2zy\=Z]^`^=�y\`Z���u�|��9\=~�_b�y^���_A�g�Lux~�_9aS~�a�u`wyZ�zy���9a5ugª9�9ay�Hø:¶¥�]_b�y���^=ux�9ay�«�G_9\ uxw{ZH\x�LuxZ2�Lu©��wy~���w«|��9\`~�_b�y^r��wyZ]X[~��]�9��^©�9\`Z�z¬Z}\`X[~�u`u`Z]�¯ux_Ñ�y_;�~�a^=u`\x��u`Z]�b~��o�]_9ay�y�y~�ux^]�Y½ ^`^=�yX¤Zou`w1��u��1�Lu��2�9\=ZP�5��u`wyZ]\=Z]�v��wy~���Z u`wyZPzy\`_A�]Z}^`^�~�^�]_9a?u`\`_b����Z}��¡5���S^=u`\x��u`Z]�9�£�M~�a[��wy~���w�Z]�9��w�ø:¶©~�^��yZ4uxZ}\`X[~�ayZ}��¡5�¸cG~QjYX¤_9ay~��ux_9\`~�a{�Su`wy\`Z}Zozy\`Z4|A~�_b�y^¥|��9\`~��9¡y��Z}^ cGø:¶*º>����ú0¶Lªy�9a{�vú·¶*º>�=j4ª��9ay�æcD~�~¦j���wy_A_b^`~�ay�

Page 16: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

dbd�fyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Êø:¶1! ¶���~�u`w¤zy\=_b¡1�9¡{~���~�ut�·�¤c ¶s� ¶*º>�:�:/ ¶;�:/ ¶Qº>�=j4�ÆÞ�w{Z zUZ]\e�G_b\`X¤�9ay�}Zr_�����~�^X[_bay~�u`_b\`Z}�«��ay�¸^`�{X¤X¤�9\`~�ä]Z}�«~�aµuxwyZS�G_b\`X�_��¥�P³=_b~�a?u zy\=_b¡1�9¡y~���~�ut���G�yay�4ux~�_ba�¤c����:/5�:�:/�0�ÀÀÀi�:/ ä � ��� ��ÀÀ0À0� >ä j4ª?��wyZ]\=ZPù�~�^��9a�_b��ux�]_9X¤Z�|��9\`~��9¡y��Z�cGZ9�ý�y��ªuxw{Zo¢?�1�9��~�ut�F_9�°uxw{Z ®1a1���¬zy\=_��{�y�}u�j4�£è�~�a1�9�����bª5��Z}u��{^��9^=^`�yX[ZFcG�G_9\�^`~�X¤z{��~��}~�ut�{juxwy��uFuxwyZµ^=ux��uxZ«ú0¶¯_9� uxwyZµzy\=_��}Z]^`^¤�yZ]zUZ]a{�y^�_ba{����_ba¼uxwyZµzy\=Z}|A~�_b�{^¤^eu���u`Zú·¶*º>���9ay�v_bavu`wyZPzy\`Z4|A~�_b�y^��}_ba5ux\=_b�¬ø:¶*º>���Æå�Zo��~�^=wvux_7Z}|��9���1��u`Z©uxwyZ©X¤Z}\`~�u_9��\`Z]z{�¦�9�}~�ay�^�'��~�uxw¤�PayZ}�Ý^eux\`��uxZ}�9�bª[� ® ª?~�a7��wy~���w[ø:¶�~�^£��wy_b^=Z]av�9�}�]_b\=�y~�a{�ux_[�FayZ}�×�]_ba{�y~�u`~�_ba1����zy\=_b¡1�9¡y~���~�ut�¸� ® c ¶�� ¶*º>���:/ ¶;��/ ¶*º>�=j4�ã��9^=Z]��_baæ_b�y\ zy\`Z4|A~�_b�y^���a1�9���A^=~�^7cGZ]¢?�1��u`~�_ba�cÀÁ{��dgÃ?j=j4ª¬u`wyZ2zUZ]\e�G_b\`X¤�9ay�}Z� ® c��{j�_9��uxwyZ�ayZ4�þ^=u`\x��u`Z]�9��� ® ��~�����¡UZ��b_�|bZ]\=ayZ]�¨¡5�vuxwyZ��y~�^=ux\=~�¡y��ux~�_9a

� ® c-�^�:/5���:/�2�ÀÀ0Ài�:/ ä � ��� �0�ÀÀÀi� >ä j cÀÁ{��d;ìbj! � ® c����</5���:/�2�ÀÀÀ��:/ ä � ��� �2�À0ÀÀi� >ä j6 � ¶ � ® c4/ ¶�� / ¶*º>��� ¶Qº>�=j � ¶ � ® c ¶�� ¶Qº>���:/ ¶;��/ ¶*º>�=j�À

ã�Z]�]�9�y^=Z�uxwyZ[®1\`^euHut��_«uxZ}\`X[^H\=Z]X¤�9~�a'~�a5|��9\`~��9a5u��9ay��u`wyZ¤u`wy~�\=�M_ba{Z¤~�^´?ay_���a/ª1��ZSw1�g|bZ

� ® c-�yjç! #$ &>»½¼½¼½¼ » $¿¾ » 7 &�»½¼½¼½¼ » 7 ¾V� ® c����:/5���:/�2�ÀÀÀ��:/ ä � ��� �2�À0ÀÀi� >ä j! #$ &>»½¼½¼½¼ » $¿¾ » 7 &�»½¼½¼½¼ » 7 ¾V�¤c����</5���:/��ÀÀ0À0�:/ ä � ��� ��0ÀÀÀ0� >ä j

6 å ¶ �¤c4/ ¶s�</ ¶*º>�:� ¶*º>�=j>å ¶ � ® c ¶�� ¶*º>���:/ ¶5�:/ ¶*º>�=j�ÀµcÀÁ{��d�h5jsta�uxw{Z ^=z¬Z}�]~¦���U�]�9^`Zr��wyZ]\=ZX� ® ~�^��{Z}uxZ}\`X[~�ay~�^=u`~����9ay�[ux~�X¤Z��t~�a5|��9\=~¦�9a5ugªLø:¶

¡UZ]�]_9X¤Z}^ �2�G�ya{�}ux~�_ba¯_9��øT¶Qº>���oú·¶Lª±�9ay�¨ú0¶Qº>��ÿ ¶j!ÁÀ/c ¶*º>���:/ ¶_�:/ ¶*º>�ej�À

Þ�wyZ}a¨uxw{ZS^=�yX[X���u`~�_ba�_;|9Z]\ ���ÀÀÀi� >ä �]�9a¨¡UZ�z¬Z}\=�G_b\=X¤Z}�/ª{�A~�Z}���y~�ay�� ® c��{jç! #$ &>»½¼½¼½¼ » $ ¾ �¤c����</5���:/��À0ÀÀ0�:/ ä �FÀs���FÀ5�0�À0ÀÀ0�FÀ ä j

6 å ¶ ��c�/ ¶�� / ¶*º>���FÀ¶*º>�ej�� cÀÁ{��dG�bj��wyZ}\`ZgÀ3¶P~�^��yZ}®1a{Z]�«\=Z]�}�y\`^=~�|bZ}���¯��^À���!ÃÀ�c4/5�ej��9ay��À3¶1!ÁÀ/cÄÀ3¶Qº>�:�:/ ¶5�:/ ¶*º>�=j�À

Page 17: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ\É{ÎÍÓOÖ7É$Ê£ËAÊ�Ê£ÖFÉ�Ó4Ù�ÖÓOÖըǩËjËHÙ?Ê$ÓtÇoÖ*Õ,Ù�ÒPÊ¥Ñ�Ð dbdg÷sta¤u`wyZ ^=z¬Z}�]~��9�1�g�9^=Z _��°��^=ux\`��uxZ}�9�7ø ® �]_bX[z¬_9^`Z]�[_9��Z]��Z}X¤Z}a5u��9\e�F�9�}u`~�_ba{^êsë{cQø:¶Ñ! ¶gj4ª$u`wyZ¤�G�{ay�}u`~�_baÅÀæ�yZ]�9Z]ayZ}\x��u`Z]^2~�a?u`_¸�µ�]_bay^eu��9a5u]ª ¶9ª��9ay�M��Z_b¡{ux�9~�a

� ® c-�yjç! �¤c������ ����� �2�ÀÀ0Ài��� >ä j! #$'& »½¼½» $ ¾ ��c-�.� /5�:�:/�2�0ÀÀÀ��:/ ä � ��� �2�ÀÀÀ�� >ä j>å ¶ �¤c4/ ¶s�</ ¶*º>�:� ¶*º>�=jcÀÁ{��d�f5j

��wy~���w¨�]�9a«����^`_F¡UZP_b¡{ux�9~�ayZ}�v�G\`_bX cDÁ{��dgÃ?j4�Þ�wyZ+zy���9ayay~�ay�×z{\`_b¡y��Z]X ~������y^eux\`��uxZ}�*¡5�uxwy~�^�Z��y��X¤zy��Zã~�^'ut�Azy~��g�9�[_9�

§«�9\=´9_�|µ�yZ]�}~�^=~�_baµzy\`_A�]Z}^`^=Z]^7cÀ§�²O´Y^�jSc4³r_;���9\=�Md�÷bh9i{Ûw²oZg�9a«�9ay�¯åMZ}����X��9ad�÷b÷�dbÛ�ã�Z}\=u`^`Z]´L�9^$�9ay��Þ�^`~�ux^`~�´?��~�^�d�÷b÷bh5j�ªL��w{Z]\`Z£uxw{Z¥ux�9\`�bZ4u�_9�1�9ay�9���A^`~�^°~�^°®1ay�A�~�a{��uxwyZ7¡UZ]^eu�a{Z4�AuS�9�}u`~�_baÄêsë�cGø:¶è! ¶gj4ª°�b~�|bZ}a�u`wyZ7�]�{\`\`Z}a5u�^eu���u`Z¤ú0¶7��ay�z1�9^eu �9�}u`~�_bay^}�Ysta¨§�²O´Y^]ªy��ZH�9\=ZPay_b\=X��9�����[�b~�|bZ]a¯uxw{Zoux\x��ay^`~�ux~�_9a��G�yay�}u`~�_ba{^�¤c4/ ¶9Æ^�2� / ¶5��� ¶�j©�9a{�¸uxwyZ2�}_b^=uP�G�{ay�}u`~�_ba«u`_v¡¬ZHX¤~�ay~�X[~�ä}Z]�/�rstaæuxwyZ2z{\`_b¡y��Z]X��Z«w1�g|bZ7³=�y^=u��9a1�����Aä]Z}�/ª�ayZ}~�uxw{Z]\F�G�yay�}u`~�_ba¼~�^F�b~�|bZ}a/Û ~�ay^=u`Zg�9��ª�uxw{Z}�¼X2�y^eu¡UZæ��Zg��\`ayZ}�+�G\`_bX �1��ux�,�5��uxw{Z]\`Z}�Ý�yay�yZ}\¯z1�9^=u�cDzy\=Z]^`�{X��9¡{���Í^`�y¡U_bz{u`~�X¤�9�Qj^=u`\x��u`Z]�b~�Z]^}�¨è1_b\eux�yay��uxZ}���bª�¡UZ]�g���y^`Z��9���Y|��9\`~��9¡y��Z]^�~�a�uxwyZ¤X¤_A�yZ}�Æ��Z}\`Z�X[Zg�L�^`�{\`Z]��ªy¡¬_�uxw��G�yay�4ux~�_bay^���Z]\=ZP~��yZ}a?u`~�®±��¡y��Z©�9ay�v�]_9�y����¡UZoZ}^=ux~�X��LuxZ]�v�{~�\`Z}�}u`����G\`_9X�uxw{ZS�}_b\`\=Z]^=z¬_ba{�y~�ay�[�]_9ay�y~�u`~�_bay�9��zy\=_b¡1�9¡{~���~�ux~�Z]^���^��G_b����_;��^}ÿ

��c�/ ¶9Æ^�2� / ¶5��� ¶gj�!��¤c4/ ¶9Æ^�2� / ¶5� ¶�j4Û��c-�.� /5�:�:/�0�ÀÀÀi�:/ ä ��� ����� ��ÀÀ0À0��� >ä j�!��¤c����</5�:��/�2�ÀÀÀi�:/ ä � ��� �2�ÀÀ0À0� >ä j�Àsta · wy�9z{uxZ}\$Ã�cÀ¿�Z]�4ux~�_9a�Ãy�ýÃ?j���Z¥��~����b�yZ]�9�5��~�u`w�z1��\=ux~��9�����r_b¡y^`Z}\=|��9¡y��Z�§¨�9\`´�_;|�yZ}�]~�^=~�_ba'zy\=_��}Z]^`^=Z]^¯c�´¥²S§â²O´£^xj4ª���wyZ]\=Z�^`_9X¤ZF^=ux��uxZ}^7ú·¶[�9\`ZF�yay_9¡y^`Z}\=|bZ}�/Û��Z]�9\`a{~�ay�ruxwyZ�u`\x�9ay^=~�u`~�_baH�9ay�F�]_b^euÆ�G�yay�4ux~�_bay^Æ~�a2uxwy_b^=Z�zy\=_b¡y��Z}X¤^$��~����A\`Z}¢?�y~�\=Z�7X¤_9\`ZP~�a5u`\`~��]��uxZPX[Z}u`wy_A�µ_���~��yZ]a5ux~�®1�g�Lux~�_9a/�

sâu�~�^°��_b\euxw2a{_9ux~�ay��uxw1��u]ª9~�aSu`wy~�^�Z4�{�9X[zy��Z9ª;ux_©z{\`Z]�{~��}u�u`wyZ�Z4­�Z}�}u$_9�±��ayZ}�^=u`\x��u`Z]�9�¤~�u¥~�^�ayZ}�]Z]^=^x�9\e��®1\=^=u�u`_HX[Zg�9^=�y\`Z |9��\`~¦��¡y��Z}^©cÀú0¶gjYu`w1��u��9\`Z©��­�Z}�}u`Z]�¡5�¼^`_bX[Z¨�}_ba5ux\=_b��|9��\`~¦��¡y��Z}^«cQø:¶*º>�=j4�!¿A�y��w+X[Zg�9^=�y\`Z}X¤Z}a?u`^��9\=Z¨�9Z]ayZ}\x�9�����^`w?�ya{ayZ]��~�a'u`wyZ¤�}�¦�9^=^`~��g�9�Y��~�uxZ}\x��u`�y\`Z7_9aMZ4��zUZ]\=~�X[Z]a5u��9�Y�{Z]^`~��baÝc · _g�ãdg÷5ì9f{ªz/�yÃbf5j�¡¬Z}�g�9�{^`Z�uxwyZ4�v��~�Z©_ba¯uxwyZP�]�9�y^`�9�°z1��u`w?���g�A^ ¡UZ}ut��Z]Z}a«ux\=Zg��u`X¤Z}a?ur��ay�_b�{u`�]_bX[Z2��ay�¸uxw?�y^ou`Z]ay�¸u`_��]_ba��G_b�yay�¸u`wyZ2�yZ}^`~�\=Z]��Z}­¬Z]�4uPZ]^eux~�X���u`Z9�O³r_����Z}|9Z]\}ªU_9�y\��9a1�9���A^`~�^�^=wy_���^ru`w1��u]ª{��wyZ]a¨zy\`_9z¬Z}\`����zy\`_A�]Z}^`^=Z]�/ª±^`�{��w«X[Zg�9^=�y\`Z��X[Z]a5ux^PX¤�g�«¡UZ7~�ay�{~�^`zUZ]a{^x�9¡y��Z2~�aæz{\`Z]�{~��}u`~�ay�¤uxwyZ7Z4­�Z}�}uS_9�¥�]Z}\=u���~�aæ�]_9a?u`\`_b�zy\=_b�b\x��X¤^}��Þ�wy~�^���~�����¡¬ZSZ]^=z¬Z}�]~¦�������¤ux\=�yZ�~�a¨^=Z]X[~��O§«�9\`´�_�|�~��9a¯X¤_A�yZ}��^ScD~À�ÂZ���ª

Page 18: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d � iyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�ʲ�½1��^r~�a5|b_9��|A~�ay�[�yayX[Zg��^`�y\=Z]�«|��9\=~¦�9¡y��Z]^xj4ª1��wy~���w«�9\`Z��9a1�����Aä]Z}�¸~�a«¿�Z]�4ux~�_baÁ{�ÂÁ{� � �Ç &)�È�É%¤��ÊÞ�wyZ�~�X¤X[Z]�{~¦��u`Z7~�X[zy��~��g��u`~�_ba¸_���uxwyZ��9a1�9���A^`~�^Szy\=_;|A~��yZ]�,~�a�u`wy~�^�^`Z]�4ux~�_baM~�^uxwy��ue»Ý�b~�|9Z]a,�µ�g�9�{^x�9�£�y~��9�b\x��X�~�a'��wy~���wM�9���Y�y~�\`Z}�}u��g�9�y^=Z]^¯cG~J�ýZ9��z1�9\=Z]a5ux^xj_9��~�a5uxZ}\=|9Z]ayZ}��|��9\`~��9¡y��Z]^P�9\=ZF_b¡y^`Z}\=|��9¡y��Z�»Ý_bayZ7�g��a�~�a{�GZ]\ozU_b^eux~�a5u`Z]\=|9Z]a5ux~�_ba�y~�^=ux\=~�¡y��ux~�_9ay^��G\`_bX�z{\`Z]~�a5uxZ]\e|bZ}a?u`~�_ba«�y~�^eux\`~�¡y�{u`~�_bay^}Û1wyZ]a{�]Z9ª¬�yay�yZ}\©^=�y��wæ�9^O�^`�{X¤z{u`~�_ba{^���Z'�g�9a!Z]^eux~�X���u`Z«uxw{Z'Z4­�Z}�}u`^¨_9��~�a5u`Z]\=|9Z]a5ux~�_bay^¯�G\`_bX z1�9^=^`~�|9ZcD~À�ÂZ���ay_bayZ��{zUZ]\=~�X[Z]a5u����Qjo_b¡y^=Z]\=|���u`~�_ba{^]ª��{^`~�a{�¨uxw{Z¤ux\=�yay�]��uxZ}�M�D�9�}u`_b\`~�äg��u`~�_ba_9� cDÁ{��d]Ã?j4�-Ë¥Z}u©uxwyZ�X¤_b\=Z���wy�9����Z]ay�b~�ay�[zy\`_b¡{��Z]X ~�^ ux_¤�yZ]\=~�|9Z2�g�9�{^x�9�$Z}­¬Z]�}u`^~�a7^=~�u`�1��ux~�_bay^$��~�´�Z�è�~��b�y\=Z�Á{��dbª9��wyZ}\`Z�^`_9X¤Z�X[Z]X2¡UZ]\=^Æ_9�^�gf º �9\=Z��yay_9¡y^`Z}\=|5��9¡y��Z©��ay��^`_Hzy\`Z4|bZ]a5u�Z}^=u`~�X¤��ux~�_ba¤_�����c =º � ¼w½ º j4�Ystav¿�Z]�4ux~�_9ay^�Á{�ÂÁH��ay�vÁ{�ýÃ���Zzy\=_;|A~��yZ�^=~�X[zy��Z��b\x�9z{wy~��]�9�buxZ}^=ux^��G_b\��yZ}�]~��{~�ay����wyZ]aè��c ���� >º j/~�^�Z}^=ux~�X���¡y��ZY~�a^`�{��w¸X¤_A�yZ}��^]�zã���u ®1\=^=ur��ZSayZ}Z]�¨u`_¤�{Z}®1ayZ�X¤_9\`Zo�G_b\=X��9�����[��w1��u ~�u�X[Zg�9a{^�G_b\7�¨�g�9�y^`�9��¢?�1�9a5u`~�ut�qÌ ux_æ¡UZ�Z]^eux~�X��9¡{��Z¤�G\=_bX z1�9^`^=~�|9Zv_b¡y^`Z}\=|���u`~�_bay^}ª£�¢?�yZ]^eux~�_baµu`w1��u��D�9����^��{ay�yZ]\�uxw{Z�u`Z]��wya{~��g���/uxZ]\=X�IDTA04K1BDI Í�@`>9BÀIJ6LK1��m������Î «o¦O§.­�®�°>Ïg~V�®>°'¥�­ç¥zªÐv �0x[¡���¨�Ñsx[��­�®>°g®>°'§�¡· �9�y^`�9��¢?�1�9a5ux~�ux~�Z}^]ª1�ya{��~�´�Z�^=u��Lux~�^eux~��g�9�/zy�9\x�9X[Z}u`Z]\=^]ªU�9\`Z��yZ4®1ayZ}�¸\`Z}�¦��u`~�|bZou`_�v�g�9�{^x�9�ÆX[_A�yZ]��× �9a{�'a{_9u�\`Z}�¦��u`~�|bZ�ux_¯��³=_b~�a5uP�{~�^=u`\`~�¡y�{ux~�_ba �)Ò�c�Ó�j©_�|bZ}\uxw{Z7^`Z}uÔ�_���_b¡{^`Z]\e|bZ}�'|��9\=~¦�9¡y��Z]^}�H¿A~�ay�}Z7ay_bayZ��{zUZ]\=~�X[Z]a5u����Æ�1��ux��zy\`_�|A~��yZ}^~�a��G_b\`X¤��ux~�_ba���¡¬_b��u1�ÕÒ�c�Ó�j��9��_ba{Z9ªy��ay�¨^=~�ay�}ZP^`Z4|bZ]\`�9��X¤_A�yZ}��^��]�9a¯�bZ]ayZ}\x��u`Zuxw{ZH^x��X¤ZS�y~�^eux\=~�¡y�{u`~�_ba�ªyuxwyZ��1�9ay�bZ}\©Z��{~�^=u`^ uxwy��u u`wyZH�{Z]^`~�\`Z}�¸¢?�1�9a5ux~�ut�µ��~����ay_9u�¡UZ��y~�^=�]Z}\`ay~�¡y��ZP�yay�9X2¡y~��b�y_b�y^=���[�G\`_bX uxwyZ��1�Lu��;»ÝZ}|bZ}a¨��wyZ}a¨~�a{®1ay~�uxZ}���X¤�9a5��^`�9X[zy��Z}^Æ�9\=Z�u��9´�Z]a/��st�yZ]a5ux~�®±�9¡y~���~�ut��Z]ay^=�y\`Z}^Æuxw1��u$uxwyZ��9�y�yZ}�F�9^`^=�yX¤zA�ux~�_bay^���Z�X¤�9´9Z��9¡U_b�{uõ× cGZ9�ý�y��ª±u`wyZS�g�9�y^`�9���b\x��zyw¨_9\�u`wyZSä]Z]\=_��}_�Z4¶¤�}~�Z}a?u`^~�a¼^=u`\`�y�4ux�y\`�9�rZ]¢?�1��u`~�_bay^xj7��~�����^=�yzyzy����uxwyZ¨X¤~�^`^=~�ay��~�a��G_b\`X¤��ux~�_baM��~�uxwy_b��uZ4��zy��~��]��ux~�ay�Ý× ~�av�G�y���/�{Z}u���~��J��� �������������������������!�Ö��� 2���I�Ä�Õ%[×Õ��+¿�Ä�$ÊÕ4¥�04BD̤c�××j�840¤>9K13¸@�6LR/=UçABâ>58]C�0TE]ç{>9K1BDIQBÀ3æ6ON[>¨R�6]TA04C�×�V�ظ07:]>93¯BQ.{>9BDÌI�:�~��yZ}a5ux~�®1�9¡y��ZµIQKÍ>'@4C�>L:�:tÙ 6ONFR�6]TA04C :[I N<8$Ng6Lß�>9K13;={>9IQßx:¯6ONFR�6]TA04C :è×Ü�>9KUT·× ��N�ßx6LRoÙ 8)Ì�c4×Ü�`jz!ÚÌ�c4× �4j6�.104K�04à904ß1�ÕÒ & c�Ó�jz!¶�ÕÒ * c�Ó�j}VOÛQN26Lç�ß6984:�04ß�àL>9BÀIJ6LK{:[>9ß�07CEIQR7IQBO0`Tb8o>9KUT;=104ß�R7IQBP6LK1CE3¸>:={>9ß�BDID>9C$:�04BDZ)Ò 6ON�Ng0`>9BDç�ßx0�:

Page 19: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ\É{ÎÍÓOÖ7É$Ê£ËAÊ�Ê£ÖFÉ�Ó4Ù�ÖÓOÖըǩËjËHÙ?Ê$ÓtÇoÖ*Õ,Ù�ÒPÊ¥Ñ�Ð d � dc�6ONO�ÕÒvc�Ó{j=jFBt6¨840v0�:4BDIQR[>9BO0`Tb8Ü6�0¤TA0�Í£K�0^Ì BO6¨840P~��yZ}a?u`~�®±��¡y��ZH�G\=_bXÝZ)Ò I N̤c�×Ü�`j�!]̤c�×Ó�4j^6�.104K�04à904ßrZÕÒ & !ÞZÕÒ * Vst�yZ}a?u`~�®±��¡y~���~�ut�¸~�^�Z}^`^=Z]a5ux~��9�¥�G_9\�~�a?u`Z]�b\`��ux~�ay�¯^=ux��ux~�^=ux~��g�9�Y�y��u��McD^`�{X¤X¤�9\`~�ä]Z}�¡5�©��c�Ó�j`jS��~�uxw¼~�ay�}_bX¤z{��Z}u`Z��g���y^x�9��´?ay_�����Z}�y�bZ¯_9�,Þ5» º ßs�Æ�9^F~�uFZ]ay�9¡y��Z}^F�y^ux_¤Z]^eux~�X¤��u`ZS¢?�1��a?u`~�u`~�Z]^OÌ"�]_ba{^`~�^euxZ}a?u`���v�G\=_bX��¦�9\=�bZ�^x�9X[zy��Z]^r_9������~�uxwy_b��u^`zUZ]�}~��Q�A~�a{�¤uxw{Z��yZ4u��9~���^r_9��×Û¬uxw{ZH�bZ}ayZ]\`�9�$��wy�9\x�9�4uxZ}\`~�^eux~��]^r_9�Æuxw{Z��}�¦�9^=^XÙ^`��¶¤�}Z9�£è1_b\�uxwyZ�z{�y\`zU_b^`Z�_��±_b�y\Y�9ay�9���A^`~�^]ª9u`wyZ�¢?�1�9a5u`~�ut��Ìá_9�U~�a5u`Z]\`Z}^=uY~�^�uxw{Z�g���y^x�9�?Z4­�Z}�}uÂ�ÕÒvc-�.��� j4ª;��wy~���wS~�^��}Z]\=ux�9~�a{���o�}_bX[zy�{u���¡y��ZY�G\`_9Xº���b~�|9Z]aHX[_A�yZ]�× cD�y^=~�ay�â²oZ}®1a{~�ux~�_ba�Á{� � ��d;j©¡y�{u���wy~���w'��Z¤_��QuxZ]a�ayZ}Z]�'ux_¯�]_bX[zy�{u`Z7�G\`_bX�9a7~�ay�]_bX[zy��Z}uxZ¥^`zUZ]�}~�®1�]��ux~�_ba2_9�^×+»Ü~�a�uxwyZ��G_9\`X�_9�U�bZ]a{Z]\x���{��w1�9\`�9�}u`Z]\=~�^=u`~��}^zU_b\=u`\x�g�bZ}�2~�aSuxwyZ��b\x�9z{w·±Ü�9^`^=_��}~¦��u`Z]����~�uxw�×�$å�Z���~����buxwyZ}\`Z4�G_b\`Z��}_bay^=~��yZ}\�P�]���9^`^ßÙ _9�¬X¤_A�yZ]��^�uxwy��u£w1�g|bZ�uxw{Z��G_9����_���~�ay�©��w1��\x�9�4uxZ]\=~�^eux~��}^£~�a7�]_9X¤X[_ba/ÿcD~Qj�u`wyZ}�M^`w1�9\=Z�uxw{Z�^x�9X[Z�z1��\`Z]a5ue�t��wy~����M�D�9X[~���~�Z}^�cG~J�ýZ9��ª�uxw{Zv^x�9X[Z��]�9�y^x���

�b\`�9zyw¬±�j4Û±�9ay�cD~�~Qj¤u`wyZ}�v~�ay�y�y�}Z�z¬_b^=~�u`~�|bZ©�y~�^eux\`~�¡y�{u`~�_bay^�_bavuxwyZ�_b¡y^=Z]\e|bZ]�µ|��9\`~��9¡y��Z}^�cD~À�ÂZ���ª�¤c�Ó�jDà+ibj4�êrZ}�¦��u`~�|9Zoux_F^`�y��w¨�]���9^`^=Z]^]ª1��Z�a{_;�×w1�g|bZ�uxwyZP�G_b����_���~�a{�y��� ����������������������âáo!$#%'&)(*%'+ÄãF,/.0 213�TÖ��� 2���I�Ä�Õ%¤×)��+¿�Ä�$ÊÕ4

-/.10¤�g�9�{^x�9� Z}­¬Z]�}u,6ONFø 6LK¼ù I�:¯:]>9IDT,BO6,840¤~��yZ}a5ux~�®1�9¡y��Z[N�ßx6LR >Ó�9ß=>Q=1.± I N¤BG.10�E]çy>9K1BDIQBD3â�¤c������ j¨@`>9K+840µ@�6LR/=Uç�Bt0`Tæç�K1I�E]ç104CE32N�ßx6LR >9K13¢=16�:4IQBDIQà90=Ußx698�>58]IQCEIQBÀ3�6ON�BQ.10µ6984:�04ß4à90`T¸àL>9ß�ID>58]C�0�:�ä¼BQ.{>9B I�:98oI N �)Ò & c-�.�4� jè! �ÕÒ * c������ jNg6Lßµ04à904ß�3;={>9IQßµ6ON¤R�6]TA04C :è×Ü�¯>9KUT�× �A6YIQBQ. �)Ò & c�Ó�jè! �ÕÒ * c�Ó�j:à�i¼>9KUT±¤c�×Ü�`j�!̱¤c�×Ó�4j�!̱�VÞ�wyZ�~��yZ]a5ux~�®±�9¡y~���~�ut�P_9���¤c������ j�Z]ay^=�y\`Z}^Æuxw1��uÆ~�uÆ~�^�zU_b^`^=~�¡y��Z�u`_o~�a{�GZ}\$u`wyZ�Z}­¬Z]�4u_9���9�}u`~�_ba¬êsë�cGø ! j�_ba¨ù��G\`_bX�ut��_[^`_b�{\`�]Z}^ _���~�a{�G_b\`X¤��u`~�_ba/ÿcD~Qj�z1��^`^`~�|bZ7_b¡{^`Z]\e|���ux~�_bay^]ª���^o^`�yX[X¤�9\`~�ä]Z]�«¡?�¨uxwyZ7z{\`_b¡1��¡y~���~�ut�v�G�yay�4ux~�_ba�¤c�Ó�j4Ûy�9ay�cD~�~Qj¤u`wyZ��]�9�y^x���A�9\x�9zywè±[ªL��wy~���wH^`zUZ]�]~�®1Z]^�cD¢?�1�9��~�ux��ux~�|bZ]���{j¬��wy~���wH|��9\=~¦�9¡{��Z]^

X¤�9´9Z7�{z'u`wyZ7^=ux�9¡y��Z7X[Z]��wy�9ay~�^=X¤^P~�a¸uxwyZF�y_9X��9~�a¸_b\}ª��9��u`Z]\`ay��ux~�|bZ]���bª��wy~���w¨|��9\=~¦�9¡y��Z]^�z1��\=ux~��]~�z1��uxZ©~�a¯uxwyZS�yZ}u`Z]\=X¤~�a1��ux~�_bav_9��Zg�9��wµ|9��\`~¦��¡y��Z~�a¯u`wyZ��y_bX¤�9~�a��

Page 20: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d ��� ÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�ÊêrZ}^=u`\`~��4ux~�ay�æ~��yZ}a5ux~�®1�9¡y~���~�ut��ux_�zU_b^`~�ux~�|bZv�y~�^=ux\=~�¡y��ux~�_9ay^2�9^=^`�y\=Z]^[�y^7uxw1�Lu

uxw{Z'�}_bay�y~�ux~�_ba+ø ! ~�^v\=Z]zy\=Z]^=Z]a5uxZ}�º~�aÝuxw{Z'�y��u���~�aÝu`wyZ'�9zyzy\=_bzy\`~���uxZ�]_9a?u`Z4�Augª�uxw?�y^S�g|b_b~��y~�a{�¨�¯ä]Z}\`_µ�yZ]a{_bX¤~�a1��u`_b\o~�aãcÀÁ���d�ibj4�7sâu���_9�y���'¡¬ZF~�XF�zU_b^`^=~�¡y��Z�ux_�~�a{�GZ}\ uxw{ZHZ}­¬Z]�4u�_9�¥�9�4ux~�_9aÇêsë{cQø ! j��G\=_bX&�y��u���~�a¨��wy~���w¸øayZ4|bZ]\o��u`ux�9~�ay^�u`wyZ�|��9���yZ ~�a¯uxwyZ��]_ba5uxZ��Auo��wyZ]\=Z]~�a¨uxw{Z2�9�4ux~�_9a¨~�^©�9z{zy��~�Z]�/�W��AuxZ]a{^`~�_9ay^Su`_¯^`_bX[Z7ay_bayzU_b^=~�ux~�|bZ7�y~�^=u`\`~�¡{�{ux~�_bay^P�9\=Z7�GZg�9^=~�¡y��Z2¡y�{u���~�����ay_�u¡UZru`\`Zg�LuxZ]�[wyZ}\`Z9�£ér_9uxZ�uxwy��ugª5ux_Szy\=_;|9Z©ay_9ay~��yZ}a5ux~�®1�9¡y~���~�ut�9ª9~�u¥~�^¥^=�{¶[�]~�Z]a5u�u`_zy\=Z]^`Z}a5u©ut��_�^`Z4ux^©_��Æ^=u`\`�y�4ux�y\`�9��Z]¢?�1��u`~�_bay^�uxwy��ur~�ay�y�{�]ZS~��yZ}a?u`~��]�9�°�y~�^eux\=~�¡y���ux~�_bay^�_�|bZ}\ _9¡y^`Z}\=|bZ}�¨|��9\=~¦�9¡y��Z]^�¡y��u�w1�g|bZS�y~�­¬Z]\=Z]a5ur�g�9�y^`�9�°Z}­¬Z]�4ux^}�å ^=~�ay��u`wyZ2�}_bay�]Z}z{u�_9�£~��yZ]a5ux~�®±�9¡y~���~�ut�bªy��Z2�g�9aæa{_;�^=�yX¤X¤�9\=~�ä]Z�uxw{Z2\`Z��^`�{��ux^�_9�Æ¿�Z}�}ux~�_baµÁ{� � � Á7~�a¯u`wyZP�G_b����_���~�ay�Hu`wyZ]_9\`Z]Xµ���� ����b 2� ��������æ 5 IQà904K¯>7@`>9ç?:]>9CyT9ID>2�9ß`>9Ṟ�6ON�>9K13[Ú�>9ß�Û56Là�ID>9K¤R�6]TA04C{IQK6�.AIJ@�.«>¤:4çy84:�04BÕÔ 6ON�àL>9ß�ID>58]C�0�:H>9ßx0SR�0`>L:4ç�ßx0`Tb8�BG.107@`>9ç5:]>9CÆ0�¦o0�@4B0ç�c-�.��� jSI�:IDTA04K1BDI Í�>58]C�0D6�.104K�04à904ßzÞ�øKarù£a1�gfjÏ1ß�èÃÔ`8�BG.{>9B°I�:98�6�.104K�04à904ß$ø���ù;8$>9KUT>9CQC�={>9ßx04K1BQ:v6ON7àL>9ß�ID>58]C�0�:FIQK«ø >9ßx0FR�0`>L:4çAßx0`T5V¼-/.10�0=<*=Ußx0�:�:4IJ6LK¼6ON(�¤c������ jI�:SBQ.104K�698]Bâ>9IQK�0`T�8]3�>5T�7�ç5:4BÀIQK�� Ng6LßA�gf 7 8�>L:�IQK'cÀÁ{��d�Á5j}V½ ^=z¬Z}�]~¦���Æ�g�9^=Z7_9�¥Þ�wyZ]_9\`Z]X�Á{� � � ì¤w{_b���y^ ��wyZ]aã>9CQC1|9��\`~¦��¡y��Z}^o�9\=Z7�9^`^=�yX[Z]�ux_F¡UZ�_b¡y^`Z}\=|9Z]�/�#���b��+�+�%[�IÊK��������é 5 IQà904KvBG.10�@`>9ç5:]>9CUT9ID>2�9ß`>9R ±�6ON©>9K13(Úæ>9ß�Û56Là�ID>9KvR�6]TA04CIQKÅ6�.AIJ@�.�>9CQC$àL>9ß4ID>58]C�0�:2>9ßx07R�0`>L:4çAßx0`Tb8�BQ.10�@`>9ç5:]>9CY0�¦o0�@4B���c-�.��� j2I�:7IDTA04K1BDIÙØÍ�>58]C�0YNg6Lß204à904ß�3¤B�6�67:4ç{84:�04BG:26ON�àL>9ß�ID>58]C�0�:�ø >9KUTHù�>9KUT�I�:2698]BJ>9IQK�0`TrN�ßx6LRBQ.10SBÀß�çAK�@`>9BO0`T N}>A@4Bt6Lß�ICB�>9BÀIJ6LK�6ON�cDÁ{��dgÃ?j]Vå�ZYay_��Mux�{\`a�_b�y\���u=uxZ}a?u`~�_baou`_�~��yZ}a?u`~�®1�]��ux~�_baozy\=_b¡y��Z]X[^°~�aP^`Z}X¤~ �e§«��\`´9_�|A~¦��aX[_��yZ}��^}�

í¯î�í ê ó ñ�ò lYóÝ�� ögñÐë ê ó ñ nbó[õ ñ ô ögñÐë ì�ö p �åÜwyZ}ayZ}|9Z]\v��Zæ�{ay�yZ]\eu��9´�Zæux_�Z}|��9���1��u`Z¨uxw{Z¸Z}­¬Z]�4u¯_9�S_bayZ«�D�9�4ux_b\¸cGø'j[_ba�9ay_�uxwyZ}\µcÀù2j�ªYuxwyZv¢?�yZ}^=ux~�_ba¼�9\`~�^`Z}^F�9^2ux_æ��wyZ4uxwyZ}\F��Zv^`w{_b�y���¼�9�;³=�y^=uF_b�{\X[Zg�9^=�y\`Z}X¤Z}a?u`^7�G_b\7z¬_9^`^`~�¡y��Z�|��9\=~¦��u`~�_ba{^H~�a¾^=_bX¤Zv_�uxwyZ}\F�D���}ux_9\`^µcJú©j4ª¥_9uxw��Z]\e��~�^=Z2´?ay_���a���^�à=�]_�|��9\=~¦��u`Z]^]ªdáÜà=�}_bay�}_bX¤~�u��9a5u`^]ªdáv_b\ à=�}_ba{�G_b�{ay�yZ]\=^Zá'c · _��d�÷5ì�f{ª±z/�±Ã5f5j4��½ �L³=�y^euxX[Z]a5uP�9X[_b�ya5u`^ru`_�z1�9\eux~�ux~�_9ay~�ay�2u`wyZHzU_bzy�{�¦��u`~�_ba¯~�a5u`_�b\=_b�yzy^Suxw1��u2�9\`Z[wy_bX[_b�bZ}ayZ]_9�y^H\=Z]�¦�Lux~�|9Z[u`_�ú�ª��9^`^=Z]^`^=~�ay�¸u`wyZ�Z4­�Z}�}u7_���ø

Page 21: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ ÌAκÄ�Ù�Ö7É$ËPÙ�Ñ�Ñ°ÓOÖHÔ"Ä�Ù�֠إÙ[Ï�Ö©ÒoÓOÖHÔÞí�ÓtÇPÐ d � Á_baMù�~�a'Z]�9��w,wy_9X¤_b�9Z]ayZ}_b�y^S�b\`_b�{z/ª���ay��u`wyZ]aM��|9Z]\`�9�b~�a{�¨uxw{Z¤\=Z]^`�{��ux^�cÀ��^~�aÝcÀÁ{��d�Á5j=j4�æÞ�wyZv~������{^`~�|9Z¤ay��ux�y\=Z�_9�r^`�{��wÍ���L³=�y^=u`X¤Z}a5u7���9^H\`Z]�}_b�bay~�ä]Z}�Í��^Zg��\`�����9^�d�fb÷b÷�ª{��wyZ]a}u��9\`�E´$Zg��\`^`_9a«�y~�^`�]_�|9Z]\`Z}�¨��w1�Lu ~�^�ay_��þ�g������Z}�Ðî±IQR/=�Ø:�6LKdï :�={>9ß=>5TA6}<©cG^`Z}Z�¿�Z}�}u`~�_ba�h{��d;j4ÿ°½ a?��^eu���u`~�^eux~��]�9�b\=Z]�¦�Lux~�_9ay^`wy~�zP¡¬Z4ut��Z}Z]aHut��_|��9\`~��9¡y��Z]^�X¤�g��¡UZP\`Z4|bZ]\=^`Z}�«¡5��~�ay�]���y�y~�a{�7�9�y�y~�ux~�_ba1�9�±�D�9�}u`_b\`^�~�a�uxw{Z��9a1�9���?�^`~�^]�rè1_b\�Z4�{�9X[zy��Z�ªU��ZHX¤�g�v®1ay�«u`w1��u ^=u`�y�yZ]a5u`^o��wy_�^=X¤_b´�ZH_b¡�u��9~�a¨wy~��bwyZ}\�b\`�9�yZ]^�uxw1�9a�uxwy_b^=Z¤��wy_æ�{_¸ay_9uH^`X[_b´9Z�¡{�{ugªY���L³=�y^=u`~�ay�µ�G_b\2���bZ9ªÆ^=X¤_9´9Z]\=^_b¡{ux�9~�aH��_���Z}\Y�b\x���yZ]^Æ~�a7Z}|9Z]\=�7�9�bZ��b\=_b�yzF�9a{�/ªb�G�y\=u`wyZ]\Æ�9�L³=�y^eux~�a{�o�G_b\$�D�9X[~����~�a{�]_bX[Z9ª±^`X[_b´�Z]\`^©�9�5�9~�a¯_b¡{u���~�aµwy~��bw{Z]\r�b\x�9�{Z]^�uxw1�9a«ay_bay^=X¤_b´�Z]\=^©~�aµZ}|9Z]\e�~�a{�]_bX[Z4�t�9�bZo�b\=_b�yz/ªy�9ay�µ^`_[_ba/�²oZ]^=zy~�u`Zo���}Z]a5ux�{\=�¤_9���9a1�����A^`~�^]ª{¿�~�X[zy^`_9a/ûÂ^¥\`Z4|bZ}\`^x�����]_9a?u`~�a?�yZ}^�ux_Çàeux\`�9zuxw{ZH�ya5���9\=��á�c4²P�g��~���d�÷o�9÷bj4ªU�9ay�¸u`wyZHzy\`�9�}u`~��]�9��¢A�{Z]^=u`~�_ba¸u`w1��u©~�urz¬_9^`Z]^t»��wyZ4uxwyZ}\v�9a+���L³=�y^=u`X¤Z}a5u��G_b\¤���b~�|bZ}a+�]_�|��9\`~���uxZµ~�^¤�9zyz{\`_bzy\=~¦��u`Z�»Ýw1�9^F\`Z��^`~�^=u`Z]�¸X¤��uxw{Z]X¤��ux~��g�9��u`\`Z]��uxX[Z]a5ug�rWYzy~��yZ]X[~�_b��_b�b~�^=ux^}ª{�G_b\rZ4�{�9X[zy��Z�ª���\`ZS^=ux~�����yZ}¡1��ux~�ay�¤u`wyZSX¤Z]�9ay~�a{�¤_9�Oà=�]_9a{�G_b�yay�{~�ay��á«c��P\x�g�A^`_9a,d�÷bfo�AÛ�¿Aw1�9zy~�\`_µd�÷b÷o�bj�9ay�«_9�QuxZ}aæ�9�L³=�y^eu©�G_b\r��\`_ba{��^`Z4ux^o_9�Æ�]_�|��9\`~���uxZ}^7cGå�Z]~�a?¡¬Z}\`�¸d�÷9÷bÁ{ÛU^`Z]Z2�9��^=_· w1�9z�uxZ]\�hbj4�YÞ�wyZPzU_9uxZ}a5ux~¦�����t_9�{ux�}_bX¤Z �9a1�����A^`Z}^�_9�$êr_b^=Z]a?¡1�9�{X��9ay�µê��y¡y~�acedg÷bfbÁ5jF�9ay�ôY\`��u`u��9ay�Ü¿A��wy�¦�9~��GZ]\«ced�÷9fbf5j7w1�g|bZ¸��Z]�Íu`_����}_bay�}Z]z{uvay�9X¤Z}�à=~��bay_b\`�9¡y~���~�ut�9ª\á¯��wy~���w�\=Z]�]�9^=u`^HuxwyZ¤�]_�|��9\`~���uxZ[^`Z}��Z}�}ux~�_ba,zy\=_b¡y��Z]X ~�a��]_b�yaA�uxZ}\=�D�9�4ux�1���$|b_A�g��¡y�y�¦��\=�¨¡y�{uP�D�9����^r^`wy_b\euP_9�¥zy\=_;|A~��y~�ay���¤��_9\`´L�9¡y��Z2^`_b���{ux~�_ba/�st�bay_9\x�9¡y~���~�ut�M\=Zg�9�y^}ÿÃàxú�~�^F��a+�9�yX[~�^=^`~�¡y��Z�^=Z}u¤_9�o�}_�|9��\`~¦�LuxZ]^F~��Oª��b~�|bZ}a+ú�ªuxw{Z�|9�����yZ�u`w1��uFù���_b�{����_b¡{ux�9~�a�wy�9�,ø ¡¬Z}Z]a ~�^2~�ay�yZ]zUZ]a{�yZ]a5uF_9��ø��dá¿�~�ay�]Z¯�]_9�ya5uxZ]\e�D�9�}u`�1�9��^¤��\`Zvay_9u[_b¡{^`Z]\e|��9¡y��Z�ª���ay�¾^`~�a{�]ZH³=�y�y�bX[Z]a5u`^¤��¡¬_b��u�]_9ay�y~�u`~�_bay�9�$~�ay�yZ}z¬Z}ay�yZ]a{�]ZF_9�£�]_9�ya5uxZ]\e�D�9�}u`�1�9��^��9\`Z2a{_9uP\`Z]�9�y~����«�9^`^=Z]\=ux�9¡y��Z�G\`_9X�_b\=�y~�a1��\=�¨�yay�yZ}\`^=ux�9ay�y~�ay�¯_9���g�9�y^`�9�Æzy\=_��}Z]^=^`Z]^}ª�u`wyZF¢?�yZ]^eux~�_9a'wy�9^P\`Z��X¤�9~�ayZ}�v_bzUZ]a/ÿÆåÜw1�Lu �}\`~�uxZ]\=~�_ba�^`wy_b�{���µ_bayZP�y^=Z�u`_F�yZ]�}~��yZP��wy~���w¯|��9\=~¦�9¡{��Z]^�9\=Z���zyzy\`_9zy\`~���uxZ©�G_9\��9�L³=�y^euxX[Z]a5u$ð

¿�Z]�4ux~�_ba'Á{� Á���dFzy\=Z]^`Z}a5ux^��¯�9Z]ayZ}\x�9�Y��ay�æ�G_b\`X¤�9��^=_b����ux~�_9a¸_9�¥uxwyZF�9�;³=�y^=ue�X[Z]a5u[zy\`_b¡{��Z]X¹�y^`~�ay�æuxwyZ¯�¦��ay�b�1�9�9Zv_9�©�]�9�y^`�9���b\x�9z{wy^]�¼sta+¿AZ]�}u`~�_ba¼Á{�ÂÁ{� ���ZPZ4�AuxZ}ay�µu`wy~�^�\`Z]^=�y��u�ux_7ay_ba{^=u���ay�1�9\=�µ�}_�|9��\`~¦�LuxZ]^�uxw1��u��9\`ZP�L­�Z}�}uxZ}�¯¡?�¤ø�9ay�'wyZ}ay�]Z[\`Z}¢A�{~�\`ZF^=Z}|bZ}\x�9�£^euxZ]z{^�_9���9�L³=�{^=uxX[Z]a5u]�«cDè�~�ay�9�����bª°¿AZ]�}u`~�_ba'Á{�ÂÁ{�ÂÁ~������y^eux\`��uxZ}^�uxwyZ��y^=Z�_9��u`wyZ]^=ZS�}\`~�uxZ]\=~¦�2~�a¨��a¨Z��y��X¤zy��Z9�

�m�g�m�3� ñ  [§óòÜ�·~�ô-õÂö�¥A¥��ÞvÈ�w°�®�§��w°k¥�­½©^=^`�yX[Z���Zr�9\=Zr�9~�|bZ}a��P�]�9�y^x���1�y~¦�9�9\x�9X ±[ªbu`_b�bZ}u`wyZ]\Y��~�uxwFay_bayZ���z¬Z}\`~�X[Z]aA�u����1�1��u��S_ba¤��^`�{¡y^`Z4urÔþ_9�°_b¡y^=Z]\e|bZ]�¤|9��\`~¦��¡y��Z}^¥~�aѱFªA�9ay�¤^`�yzyzU_b^=Z ��Z ��~�^=w

Page 22: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d � Ã1ÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Êux_ Z]^eux~�X¤��u`Z¥��w1�LuÆZ}­¬Z]�4uÆuxw{Z�~�a5u`Z]\=|9Z]a5ux~�_bay^Âê�ë�cGø ! j°��_9�y����w1�g|bZ�_ba2� ^`Z4u_9�°\=Z]^=z¬_ba{^`Z©|��9\=~¦�9¡y��Z]^¥ù�ª?��wyZ]\=Z©ø ��ay�vù�9\`Zrut��_2^=�y¡y^=Z}ux^�_���Ô[�Ysta�_�uxwyZ}\��_b\`�y^}ª1��Z�^`Z}Z]´vux_FZ}^=ux~�X��LuxZ(�¤c������ j��G\=_bX��7^x��X¤zy��ZPZ]^eux~�X¤��u`ZP_9���¤c�Ó�j��

åMZ�^=wy_��Íu`w1��uÆu`wyZ]\=Z�Z4��~�^eux^£�©^=~�X[zy��Z��b\x��zywy~��]�9�?uxZ]^eugª5a1��X¤Z}�2uxwyZÝà=¡y�9��´b��y_A_b\7�]\=~�u`Z]\`~�_ba�á¸~�a©´$Zg��\`�Scedg÷b÷bÁ9¡±j�ªYuxw1��uF�]�9a¼¡¬Zµ�9zyz{��~�Z}���y~�\=Z]�4ux���'ux_�uxw{Z�g���y^x�9���y~��9�b\`�9X�~�aµ_b\=�yZ]\�ux_[uxZ}^=u ~��Y�F^`Z4uPú{èÚÔ"_9��|��9\=~¦�9¡{��Z]^�~�^r^=�{¶[�]~�Z]a5u�G_b\�~��yZ]a5ux~��Q�A~�ay���¤c������ j���÷�� ����������������������� "!Føt%'1Iù�ã9�������´4ú :�04B¯6ONæàL>9ß�ID>58]C�0�:¯ú :]>9BÀI�:�Í�0�:æBG.10µ¡1����´5�â�y_A_b\¾@4ß�IQBt04ß�IJ6LKºßx04C�>9BÀIQà90'Bt6Í>9K6Lß=TA04ßx0`Tg={>9IQß26ONPàL>9ß�ID>58]C�0�:�cGø º �=ø�4j�IQK'>;û ú 5 ±�I N*ücD~Qj¸K�6FK�6]TA0PIQKµú�I�:�>vTA0�:�@�04KUT5>9K1B�6ON�ø º¿ý >9KUTcD~�~Qjvú 8]C�6�@:Û�:¨04à904ß�3¢={>9BQ.¼8404B�6�0�04K�ø º >9KUTvø¸BQ.{>9BH@�6LK1Bâ>9IQK{:¯>9K+>9ß�ßx6b6IQK1Bt6�ø º Vî±IQR7IQC�>9ß�CE3I8�I N¥ø >9KUTPù�>9ßx0oB�6�6[T9I�:�7g6LIQK1B$:4ç{84:�04BQ:�6ON K�6]TA0�:oIQK ±t8YBQ.104K�ú I�::]>9IDTPBO6 :]>9BÀI�:GN�3PBQ.10�8�>A@:Û_ØeTA6�6Lß�@4ß�IQBO04ß�IJ6LK[ß�04C�>9BDIQà90�Bt6ScQøÇ��ùHj�I N�IQB�:]>9BDI�:�Í�0�:�BQ.10@4ß�IQBO04ß�IJ6LK¸ßx04C�>9BDIQà90SBO6�>9K13X={>9IQßScGø º �`ø]j©:4ç±@�.µBQ.{>9B�ø º�þ ø >9KUTSø þ ùvVÞ�wyZ7a1��X¤ZÇà=¡1�9��´b�t�{_�_b\3á�Z]��w{_�Z}^P�]_bay�{~�ux~�_ba¾cG~�~Qj�ª1��wy~���wæ\=Z]¢?�y~�\=Z]^ou`w1��uo_9ay���z1��u`wy^Æ��~�u`wF�9\`\=_���^Yz¬_b~�a5ux~�a{�o��uÆø º ¡UZ�¡y��_���´�Z]�/Û5uxw{Z]^`Z�z1�Luxwy^Y�]�9a7¡¬Z�|A~�Z}��Z]��9^¯Z]a5u`Z]\`~�ay��ø º uxwy\=_b�y�bwÜuxwyZæ¡y�9��´Ý�y_A_b\]� staáè�~��b�y\`Z¸Á{�ÂÃ{ª �G_b\�Z��{�9X¤z{��Z9ªuxw{Zµ^=Z}u`^�ú�� !ÆÞ�øÝ�`ø � ß«�9ay�Íú���!ÆÞ�ø � �`ø ÷ ߨX¤Z}Z}u7u`wyZ¯¡1�9��´b�t�y_A_b\7�}\`~��uxZ}\`~�_ba/ª�¡{�{u7ú��! Þ�ø � ߯�y_AZ]^�ay_9u�¡¬Z}�g�9�{^`Z�ø � �{_�Z}^Hay_9uH¡y��_A��´¸uxw{Z�z1��u`wcGø º �`øÝ�`ø ���`ø � �`øÝ��=ø ÷ �=ø�4j����� ����b 2� ����������!Føt%¤1bù�ã<�������£���2� &�(*�I�e 2���34ÛGN�> :�04B�6ON¥àL>9ß�ID>58]C�0�:�ú!:]>9BDI�:�Í�0�:�BQ.10�8�>A@:Û_ØeTA6�6Lß�@4ß4IQBt04ß�IJ6LKFß�04C�>9BDIQà90¥Bt6�cQøÇ��ùHj98BQ.104K×BQ.10�@`>9ç5:]>9CS0�¦o0�@4B¤6ON¤ø 6LKãù I�:æIDTA04K1BDI Í�>58]C�0¸>9KUT¼I�: �9IQà904Kº8]3�BQ.10Ng6Lß�R7ç�C�> �¤c������ j�! #�$ �¤c���� ��/5j3�¤c4/5j�À cÀÁ{��d�÷5j

ÿ3CEDGL)W.f'a�L\J4Fka�LMI�Hõ}�K��õK�y)W�I�v;F�I�v2JgK�LMHGF3YjlSa�I�} CEDGFkIia4F3}¤ý{Mb�I�l��`GLMa4J4F3W�F'J.K�yq{�Q�b3ñ�ñi�iT'{��H K�y\J4F3a4H5K:J4LMx�F�f'a�L\J4Fka�LMI�Hs~oX_W4LMHGhõK[W4L\H_h�yMF�t�úûW4Fk`5K�agK:J4LMI�H(J4F3W�J3~�LMWÂFZW-JgK�vGyMLMW4DGFZY?LMH �2F3f�J�L\IiH�G{ �OQ8W4FkFmQ8�_{ �ý�T4T�{

Page 23: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ ÌAκÄ�Ù�Ö7É$ËPÙ�Ñ�Ñ°ÓOÖHÔ"Ä�Ù�֠إÙ[Ï�Ö©ÒoÓOÖHÔÞí�ÓtÇPÐ d � ìX1

XiX6

X4

X3

X2

X5

Xj

è�~��b�y\`ZHÁ{�ýÃyÿ�½ �y~¦���b\x�9X \=Z]zy\=Z]^`Z}a5ux~�a{��uxwyZ7¡1����´5�â�y_A_b\©�}\`~�u`Z]\=~�_ba/Û/�9�L³=�y^eux~�a{��G_b\v|��9\`~��9¡y��Z]^�Þ�øÝ2�`ø � ßÝcD_b\ÇÞ�ø � �`ø ÷ ß�j¤�A~�Z}���y^µ�¾�]_ba{^`~�^euxZ}a?u¨Z]^eux~�X���u`Zæ_9��¤c ���� >º j��Þ�wyZ ^`�yX[X��Lux~�_9a7~�a«cÀÁ{��d�÷5jÆ\`Z}zy\`Z}^`Z}a?u`^�uxwyZr^=ux�9ay�1��\`�¤�G_9\`X2�y���P_b¡{u���~�ayZ}�¤�yaA��yZ}\«�9�;³=�y^=u`X¤Z}a?uµ�G_b\¨ú�ÛS|9��\`~¦��¡y��Z}^¯ø �G_b\v��wy~���wáuxw{Z�Z}¢A�y�9��~�ut�Ý~�a�cÀÁ{��d�÷5j~�^P|��9��~��æ��Z}\`Z�ay�9X¤Z}�¶àe�]_bay�{~�ux~�_ba1�9�����¨~��ba{_b\x�9¡{��Z7�b~�|bZ]a,újá«~�aMê�_b^=Z]a?¡1�9�yX�9ay�¾ê��y¡y~�aÜcedg÷bfbÁ5j��æêrZ}�y�y�}~�ay�¸~��bay_b\`�9¡y~���~�ut�¸�]_bay�{~�ux~�_bay^�ux_¸u`wyZ��b\`�9zywy~��g�9��]\=~�u`Z]\`~�_ba¼_9�(²oZ}®yay~�u`~�_baÍÁ{�ÂÁ{��d«\`Z]z{�¦�9�}Z]^�³=�{�y�bX[Z]a5ux^��9¡¬_9�{u��}_b�ya5uxZ}\=�D�9�4ux�1����yZ}z¬Z}ay�yZ]a{�]~�Z}^S��~�uxw¤³=�y�{�bX¤Z}a5ux^��9¡U_b�{uou`wyZ7^=u`\`�y�4ux�y\=ZF_9�¥�g�9�y^`�9�Æzy\=_��}Z]^=^`Z]^}ª�9^�\`Z]z{\`Z]^=Z]a5uxZ}�¾~�a�uxwyZ¤�y~¦�9�9\x�9Xµ��Þ�wyZ��9\x�9zyw{~��g���Y�]\`~�uxZ}\`~�_9a��]�9a,¡UZ¤uxZ}^=u`Z]�¡5�M^e�A^=uxZ}X��Lux~��¤zy\`_A�]Z}�y�y\=Z]^Hu`w1��u7��\`Z���zyzy��~��g�9¡{��Z7ux_¨�y~¦�9�9\x�9X[^�_����9a5�M^=~�ä}Z�9ay�«^`w1�9zUZ9�rÞ�wyZ��]\`~�uxZ}\`~�_9a«�9��^`_¤Z]a1�9¡{��Z]^ruxwyZH�9a1�9���A^=urux_�^=Zg�9\=��w¸�G_b\ �9a¸_bz��ux~�X����¥^=Z}uF_9�r�}_;|��9\=~¦��u`Z�»Ýa1�9X[Z]���9ªÆ�¸^=Z}u[ú�u`w1��u7X[~�ay~�X¤~�ä]Z}^�X[Zg�9^=�y\`Z}X¤Z}a5u�]_9^=uF_b\7^`�9X¤z{��~�a{�«|��9\`~��9¡y~���~�ut�¼cGÞ�~¦�9a¾Z4u[�9�J��d�÷b÷bf5j���Þ�w{Zµ�{^`Z¯_9�©�¸^`~�X[~����9\�b\`�9zywy~��g�9���]\=~�uxZ}\`~�_ba«�G_b\o~��yZ]a5u`~��Q�A~�ay��z1��uxwæ�}_�Z4¶[�]~�Z}a5ux^P~�aæ��~�ayZ]�9\r^=ux\=�y�}u`�y\x���Z]¢?�1�Lux~�_9ay^©~�^r�{Z]X[_bay^=u`\x��u`Z]�¸~�a · w1��z{uxZ}\Pì���½ zyzy��~��g��u`~�_ba{^�u`_¤Z}zy~��{Z]X[~�_b��_b���~��]�9��\`Z]^=Zg�9\=��w'�9\`Z��b~�|9Z]aæ~�aÇ��\=Z]Z}ay�¦�9a{�¸Z}u����J�$ced�÷9÷b÷b�5j�ªU��wyZ}\`Z2u`wyZ2^`Z4uSú*~�^�g������Z}�©à=^`��¶¤�}~�Z}a?u ^`Z4u�áF�G_b\��]_9a?u`\`_b�°_����}_ba{�G_b�ya{�y~�ay�{�³^�b�d��¡r��¡ ��� 2���I 2� ���������Þ�wyZFzy\=_�_9��_b\=~��b~�a1�9�����¯_9­¬Z]\`Z}��~�aÓ´$Zg��\`�rcedg÷b÷bÁ9¡±j����9^S¡1�9^=Z]�'_ba�uxwyZF_b¡y^=Z]\O�|���ux~�_ba¸uxwy��ugª/��wyZ}a�ú ¡y��_���´?^��9���Æ¡1�9��´b�t�{_�_b\Pzy��uxwy^P�G\=_bX&ø ux_¯ù¤ª°^`Z}u=ux~�ay�cGø ! j�_b\��}_bay�y~�ux~�_bay~�a{�2_bavøç! w1��^�u`wyZ�^x�9X[ZPZ}­¬Z]�4ur_baµù��YÞ�wy~�^��]�9a¡UZ]^=u�¡UZ¤^=Z]Z]a'�G\`_bX&uxwyZ¤�9�y�bX[Z]a5u`Z]�M�{~¦�9�b\`�9X ± = _9��è�~��b�y\=ZFÁ{� � ª�u`_µ��w{~���wuxw{Zo~�a5uxZ}\=|9Z]a5ux~�_ba¯�9\`�}^ÜZ.Ï c ø���Z]\`Z����y�yZ]���¥sâ�$�9���±¡1�9��´b�t�y_A_b\�zy��uxwy^��G\`_bXø ux_Fù"��\`ZP¡y��_���´�Z]�/ª{u`wyZ]aµ�9����z1�Luxwy^��G\=_bX Z�Ï+ux_Fù XH�y^=u��9_2uxwy\=_b�y�bwvuxw{Z��wy~����y\=Z]a«_��Æø�ª¬�9ay�«u`wy_b^`ZS��_9�y���¸¡UZH¡{��_A��´9Z]�«~�����Z2�}_bay�y~�ux~�_9a¨_ba¨ø���Þ�wyZ

Page 24: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d � hyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê~�X[zy��~��g�Lux~�_9a¤~�^�u`w1��urù"~�^�~�ay�yZ}z¬Z}ay�yZ]a5u _9�ÕZ.ÏÜ�b~�|bZ]avø�ª

�¤c���� �9Z�Ï©!�êsë�c j`j�! ��c-�.� �9Z.Ï©!Ü~��y��Zgj�! ��c-�.� j�� cÀÁ{� � i5j��wy~���w�X¤Z]�9ay^¥uxw1��u�uxwyZ _b¡y^=Z]\=|���u`~�_ba[øç! �g�9a{ay_9u�¡UZ©�{~�^=u`~�ay�9�y~�^=wyZ]�[�G\`_bXuxw{ZS~�a5uxZ]\e|bZ}a?u`~�_ba}Z.Ï¢!�ê�ë�c j��è1_b\`X¤�9�����bª5��ZP�g�9a¯zy\=_;|9Zouxwy~�^�_b¡y^=Z]\e|9�Lux~�_9av¡?�¤��\`~�ux~�a{�·��c-�.�4� j�~�a�uxZ}\`X[^_9��uxw{Z��9�y�bX[Z]a5uxZ}�vzy\`_9¡1�9¡y~���~�ut�F�G�ya{�}ux~�_ba � = ~�a¯���]�]_9\`�1�9a{�]Zo��~�u`w'cÀÁ{�Â÷5j���ay��]_9ay�y~�u`~�_ba{~�ay�2_9a¸ú×ux_F_b¡�u��9~�a

��c-�.�4� j�!�� = c-�.� Z 7 j"! #�$Õ� = c-�.� /��9Z 7 j3� = c�/^�½Z 7 j! # $ � = c-�.� /�� �9Z 7 jk� = c4/^� Z 7 j�À cÀÁ{� � d;j

Þ�wyZ7���y�y~�u`~�_ba¨_9� ux_¤uxwyZ��¦�9^euoZ4��zy\=Z]^=^`~�_9aæ~�^©��~��}Z]ay^=Z]�¸¡5�µuxwyZ2~�X¤z{��~��]��ux~�_baZ 7 � ø ! �HÞ�_µZ}��~�X¤~�a1��uxZZ 7 �G\=_bX&u`wyZ2ut��_¯uxZ}\`X[^P_baæuxwyZ7\=~��bw5ue�tw1�9a{�^`~��yZ7_9�ocÀÁ{� � d;j�ª/��ZF~�a?|9_b´9ZFuxw{ZFut��_µ�]_bay�{~�ux~�_bay^o_���²oZ4®1ay~�u`~�_ba�Á�� Á{��db�F¿A~�ay�}ZZ 7 �}_bay^`~�^=u`^v_9�S\`_A_9uvay_A�yZ]^���~�uxwÜ��wy~����y\=Z]aÝ\`Z}^=u`\`~��4uxZ}�Üu`_,ø�ªr~�uvX2�{^=u¯¡UZ~�a{�yZ]zUZ]ay�{Z]a5u©_��Y�9���/ay_bay�{Z]^`�}Z]ay�y�9a5ux^o_9�$ø�ª1~�a{�]���{�y~�ay�[úS��Þ�w?�y^}ª¬�}_bay�y~�ux~�_9acD~¦j¥�A~�Z}���y^ � = c4/^� Z 7 j�!Ì� = c4/5jz!��¤c4/5j�Àsta5|b_b´?~�ay�©ay_��,u`wyZ¥¡1�9��´b�t�y_A_b\��]_9ay�y~�u`~�_ba¤cG~�~Qj�ª�ux_b�bZ4uxwyZ}\°��~�uxw�cÀÁ�� � ibj4ªLz¬Z}\`X[~�ux^�y^�ux_FZ}��~�X[~�ay��uxZ/Z 7 �G\`_bX�cÀÁ{� � d;j�ª{uxw?�y^rzy\`_�|A~�a{�«cDÁ{��d�÷5j4� ��m�g�m��� ñ  [§��[�>¥�­�®2õÂö�¥1¥z�ÞvÈ�w°�®�§��w°k¥�­· _bay�{~�ux~�_ba[cD~¦j�_9��²oZ}®1ay~�ux~�_ba�Á{�ÂÁ{��d£\=Zi�1Z}�}ux^�uxw{Z¥z{\`Z}|��9~���~�ay��zy\x���}ux~��]ZÆu`w1��uõàeuxw{Z�]_9ay�]_bX[~�ux�9a5uY_b¡y^=Z]\e|9�Lux~�_9ay^£^`wy_b�{���F¡¬Z�¢?�y~�uxZ��ya1�L­�Z}�}uxZ}�¤¡5�2u`wyZ�ux\=Zg��u`X¤Z}a?uZác · _g�Fdg÷5ì9f{ª�z/�]Ã5f5j4��Þ�wy~�^°^`Z}�}ux~�_ba��yZ]X[_bay^eux\`��uxZ}^�w{_;���]_9ay�]_bX[~�ux�9a5ux^/u`w1��u¥>9ßx0��­¬Z]�4uxZ]�H¡?�Suxw{Z�ux\=Zg��u`X¤Z}a5uÆ�g��a2¡¬Z¥�y^=Z]�2ux_r�D�9�]~���~�u���u`ZY�g�9�y^`�9�{~�a{�GZ]\=Z]ay�}Z9�$Þ�wyZZ]X[Z]\=�b~�a{� �}\`~�u`Z]\=~�_ba/ªLa1�9X[Z]��uxwyZ¥�G\`_ba5u=�â�y_A_b\��]\`~�uxZ}\`~�_9a�~�a ´$Z]�9\`��cedg÷b÷5ì9�5j�ªL��~�����]_9ay^=u`~�ux��uxZ�u`wyZS^`Z]�}_bay�æ¡y�{~����{~�ay�F¡y��_���´v_9��uxwyZ��bZ]ayZ}\x�9��uxZ}^=u �G_b\�~��yZ}a?u`~��Q�A~�a{��g���y^x�9�°Z4­�Z}�}u`^2cÀ¿�Z]�4ux~�_9a¨Á��ÂÃ?j��

· _bay^=~��yZ}\�uxwyZæ�y~��9�b\`�9X ~�aÜè�~��9�y\`Z«Á{� ìAªr��wy~���wÜ\`Z]z{\`Z]^=Z]a5ux^µuxwyZæX[_A�yZ]�_9�,è�~��b�{\`Z�Á{�ýà ��w{Z]a&u`wyZ�|9��\`~¦��¡y��Z}^+ø��:�ÀÀÀi�`ø ÷ �9\=Z��yay_b¡y^=Z]\e|bZ]� ��ay�Þ�ø º �`ø���=ø�߸�9\`Zv\=Z]�¦��¡¬Z}��Z]�ÃÞ�ø���új��ù·ß?ª�\=Z]^=z¬Z}�}ux~�|bZ}���b�¼½©��uxwy_9�y�bwÍú��y_�Z}^ay_9u¤^x�Lux~�^e�Q�¼�9a5�¾_9�ou`wyZ¨¡y�9��´b�t�y_A_b\[�]_bay�{~�ux~�_bay^}ª�X[Zg�9^=�y\`Z}X¤Z}a?u`^¤_��Sú��]�9a

Page 25: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ ÌAκÄ�Ù�Ö7É$ËPÙ�Ñ�Ñ°ÓOÖHÔ"Ä�Ù�֠إÙ[Ï�Ö©ÒoÓOÖHÔÞí�ÓtÇPÐ d � �ayZ4|bZ]\euxwyZ}��Z}^`^¥Z]a1�9¡{��Z��]_9ay^`~�^=uxZ}a5u�Z}^=u`~�X¤��ux~�_baH_9�^�¤c������ j���Þ�wy~�^Æ��~����{¡UZ�^`wy_���a¡5��\=Z]�y�y�}~�ay��uxwyZoZ���zy\`Z}^`^`~�_bav�G_b\��¤c������ j£ux_2�G_9\`X2�y���9^£uxw1��u��9\`Z©�]_bX[zy�{ux�9¡y��Z�G\`_9X�uxw{ZS_9¡y^`Z}\=|bZ}�¸�y~�^=ux\=~�¡y��ux~�_9a��G�yay�}u`~�_ba¬�¤c �����:/5j��

X Z Y

U (Unobserved)

è�~��b�y\`Z�Á{� ì�ÿ�½ã�y~��9�b\`�9Xº\`Z}zy\`Z}^`Z}a?u`~�ay�ou`wyZ��G\=_ba5u=�â�y_�_9\��}\`~�uxZ]\=~�_ba��$½¼ut��_��â^=u`Z]z�9�L³=�{^=uxX[Z]a5u��G_b\rú×�A~�Z}���y^r�7�]_ba{^`~�^euxZ}a?u Z]^eux~�X���u`Zo_9�Â��c-�.�4� j4�

Þ�wyZ�³=_9~�a5u��y~�^eux\=~�¡y�{u`~�_baS�9^=^`_A�]~���uxZ}�S��~�uxw�è�~��b�{\`Z¥Á{� ì��]�9aH¡UZ£�yZ]�}_bX¤zU_b^=Z]�cDZ}¢?�1��ux~�_ba�cÀÁ{� ìbj=j�~�a5ux_

��c �����:/���Ô¬j�!Ì�¤c�Ô¬jk��c � Ô¬j3�¤c4/^� j3�¤c����</��ZÔ�j�À cÀÁ{� ��� jè1\`_bX cDÁ{��d�i5j4ª1u`wyZH~�a?u`Z]\e|bZ]a5u`~�_ba�ê�ë�c j�\`Z}X¤_�|9Z]^©u`wyZ��D�9�4ux_b\j�¤c � Ô¬j��9ay�¨~�a���y�y�}Z]^�uxwyZ�zU_b^=u`~�a5uxZ}\=|9Z]a5ux~�_ba¨�{~�^=u`\`~�¡y�{ux~�_ba

�¤c����:/���Ô���� j�!��¤c����</���Ô¬jk��c�/^� jk��c-Ô�j�À cÀÁ{� � Á5j¿��yX[X[~�ay�7_�|bZ}\1/¤�9a{�âÔµuxwyZ}a¨�b~�|bZ}^

��c-�.�4� j�!¶#:$Õ�¤c4/^� j># � �¤c����</��ZÔ�jk��c-Ô¬j�À cÀÁ{� � Ã?jstaµ_b\`�yZ}\�ux_[Z]��~�X¤~�a1��u`ZõÔ«�G\=_bX�u`wyZ�\}�Âw��Â^}�U_���cDÁ{� � Ã?j4ªy��Z��{^`Z�uxwyZPut��_��]_9ay�y~��ux~�_ba1�9�¬~�ay�{Z]zUZ]ay�yZ}ay�]ZH�9^`^=�yX[z{ux~�_bay^�Z]a{�]_A�yZ]�¨~�avuxwyZ��b\`�9zyw¯_9��è�~��9�y\`Z�Á{� ì�ÿ

��c-Ô��</�� jç! ��c-Ô�� j�� cÀÁ{� � ìbj��c-�.� �:/���Ô¬j ! ��c-�.� /���Ô¬j�À cÀÁ{� � h5j

Þ�wy~�^��A~�Z]���{^�uxwyZ�Z]¢?�1����~�u`~�Z}^#���c-�.� /���Ô¬j3�¤c�Ô¬j ! # 7 #

��¤c����</��ZÔ�jk��c-Ô�� j3�¤c j

! # 7 #��¤c���� �:/���Ô¬j3�¤c�Ô�� �:/bj3�¤c j

! # 7 �¤c���� ��/5j3�¤c j cÀÁ{� � �bj

Page 26: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d � fyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê�9ay�¾������_���^SuxwyZv\=Z]�y�y�4ux~�_ba�_9�HcÀÁ{� � Ã?jPux_��«�G_b\=X ~�a5|b_b��|A~�ay�¸_9ay����_b¡y^=Z]\=|9Z]�¢?�1�9a5ux~�ux~�Z]^]ÿ �¤c������ j�! # $ ��c�/^� j #87 9 ��c-�.� >= �:/5j3�¤c >= j�À cÀÁ{� � f5j

½©�����D�9�}u`_b\`^F_ba¼uxwyZµ\]�ýw/�ý^]�£_9�FcDÁ{� � f5j7�9\`Z¯�}_bay^`~�^=u`Z]a5ux���¾Z]^=u`~�X¤�9¡y��Z��G\`_bXay_ba{Z4��z¬Z}\`~�X¤Z}a?ux�9���1�Lu��{ª¥^`_�~�uH�G_b����_;��^�uxw1��uÝ�¤c������ jH~�^7Z]^eux~�X��9¡{��Z���^F��Z]���J�Þ�w?�y^]ª$��Z��9\`ZF~�a'zU_b^`^=Z]^`^=~�_baM_9���9a�~��yZ}a5ux~�®1�9¡y��Z7a{_bayz1�9\`�9X[Z}ux\=~��FZ]^eux~�X��9a{��G_b\vuxw{Z'�]�9�y^`�9��Z}­¬Z]�}u¨_9��ø _baáù ��wyZ]ayZ4|bZ}\«��Z'�g��aá®1ay�á�¾X¤Z}�y~¦��u`~�ay�|��9\`~��9¡y��Z�ú×uxw1��urX[Z]Z4ux^�uxwyZ��}_bay�y~�ux~�_9ay^�_9��cÀÁ{� � ìbj���ay�,cDÁ{� � h5j4�

WY¢A�y��ux~�_ba�cÀÁ{� � f5j��]�9a¯¡¬ZP~�a?u`Z]\=zy\`Z4uxZ]�¨�9^��2ut��_��t^euxZ]z¨�9zyz{��~��]��ux~�_ba�_9��uxw{Z¡1�9��´b�â�y_�_9\ �G_b\=X2�y�¦����sta«uxwyZ�®1\`^euo^=u`Z]z/ªU��Z�®1ay�¸uxw{ZH�g���y^x�9��Z}­¬Z]�4uP_9�Æø _baú�Û1^=~�ay�}Z�u`wyZ]\=Z�~�^�ay_F¡1�9��´b�â�y_�_9\rzy��uxw¯�G\`_bX ø ux_¤ú�ªy��Z�^=~�X[zy���¤w1�g|bZ

��c�/^��� j�! ��c�/^� j�Àé Z��Augªy��ZS�}_bX[zy�{uxZ uxwyZ��g���y^x�9��Z}­¬Z]�}u�_��Yú×_baµù[ª{��wy~���wv��Z��g�9a¯ay_F��_9ay�bZ]\Z]¢?�1�LuxZ ��~�uxw[uxwyZr�]_ba{�y~�u`~�_ba1���1zy\`_b¡y�9¡y~���~�ut� ��c-�.� /5jÆ¡¬Z}�g�9�y^=Z u`wyZ]\=Z©~�^���¡1�9��´b��y_A_b\oz1�Luxw'ú� ø� Ò c ù��G\`_bX�ú*ux_¯ù��?³r_;��Z}|9Z]\}ª$^=~�ay�}Z2ø ¡y��_A��´?^c4ê9�â^`Z]zy�9\x��u`Z]^xj�uxwy~�^Hz1��u`w/ªYø �g�9a�z{�¦�g��u`wyZv\`_9��Z�_��©�¸�}_bay�}_bX¤~�u��9a5uH~�a,uxw{Z¡1�9��´b�â�y_�_9\ �}\`~�u`Z]\=~�_ba/ªy��w{~���wæ�9����_���^��y^�ux_��}_bX[zy�{uxZ�uxw{ZH�g���y^x�9��Z}­¬Z]�4u©_��¥ú_ba�ù�~�aM�9�]�}_b\`�1��ay�]Z[��~�uxwÝcÀÁ{��d�÷5j�ª��b~�|�~�ay�â��c-�.���/bj?!�� 7:9 �¤c���� = �:/5jk��c = j4�è�~�a1�9�����bª?��ZS�]_9X2¡y~�a{ZouxwyZPut��_¤�]�9�y^x���°Z}­¬Z]�}u`^r|A~��

��c-�.�4� j�!Ì# $ ��c-�.�4�/bjk��c�/^��� j����wy~���w¨\=Z]�y�y�}Z]^�ux_«cÀÁ{� � f5j��

åMZ�^`�yX[X¤�9\`~�ä]Z�uxwy~�^ \=Z]^`�{��u ¡?�«�[u`wyZ]_9\`Z]X&���Qu`Z]\��G_b\`X¤�9�������yZ4®1ay~�ay��uxw{Z�9^=^`�yX[z{ux~�_bay^}��� �������������������������!� ��b�����*ã9�������´4ú :�04Bo6ON2àL>9ß�ID>58]C�0�:�ú I�:H:]>9IDT¯BO6�:]>9BDI�:GN�3vBQ.10P�G\`_ba5ue�t�y_A_b\�@4ß�IQBO04ß�IJ6LK�ß�04C�>9BDIQà90Bt6�>9KM6Lß`TA04ßx0`Tg={>9IQßH6ON�àL>9ß�ID>58]C�0�:�cQøÇ��ùHjSI N*ücG~Qj¯ú�IQK1Bt04ßx@�0>=UBG:H>9CQC�T9IQßx0�@4Bt0`T^={>9BG.5:ÆN�ß�6LR*ø BO6�ù ýcG~�~¦j«BG.104ßx0SI�:SK�6¤8�>A@:Û_ØeTA6�6LßD={>9BQ.�N�ßx6LR�ø Bt6Hú ý >9KUTcG~�~�~Qj¸>9CQC�8�>A@:Û_ØeTA6�6LßD={>9BQ.5:YN�ßx6LRÚú�Bt6�ù�>9ßx0H8]C�6�@:Û50`T�8]3�ø¼V

Page 27: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ ÌAκÄ�Ù�Ö7É$ËPÙ�Ñ�Ñ°ÓOÖHÔ"Ä�Ù�֠إÙ[Ï�Ö©ÒoÓOÖHÔÞí�ÓtÇPÐ d � ÷��� ����b 2� �������âáo!� ��I������ã<�������£���2� &)(Q�I�e 2���34ÛGNYúþ:]>9BDI�:�Í�0�:�BQ.10°N�ßx6LK1B4ØeTA6�6Lß©@4ß�IQBt04ß�IJ6LK�ßx04C�>9BÀIQà90�BO6�cQøÇ��ùHj >9KUTSI N��¤c �:/5j-ài28rBQ.104K,BQ.10�@`>9ç?:]>9C¥0�¦o0�@4BP6ON©ø 6LKæù I�:FIDTA04K1BDI Í�>58]C�0[>9KUT¨I�:,�9IQà904K�8]3µBQ.10Ng6Lß�R7ç�C�> �¤c������ j�! #�$ ��c�/^� j #87 9 ��c-�.� >= �:/5j3�¤c >= j�À cÀÁ{� � ÷5j

Þ�wyZP�}_bay�y~�ux~�_9ay^�^eu���u`Z]�v~�a¸²oZ}®1ay~�ux~�_ba�Á{�ÂÁ{�ÂÁ7�9\`Z©_;|9Z]\=���¤\`Z]^eux\=~��}u`~�|9Z9Ûy^=_bX¤Z _9�uxw{Z�¡1�9��´b�â�y_�_9\�z1��u`wy^ÆZ��{�}���y�{Z]�7¡5�S�]_bay�{~�ux~�_bay^�cD~�~Qj°�9ay�¯cG~�~�~Qj��g��a7�9�}u`�1�9������¡UZ�9����_���Z}�7zy\`_�|A~��{Z]�FuxwyZ4�F�9\`Z�¡{��_A��´9Z]�[¡5�7^`_9X¤Z��}_bay�]_9X¤~�u��9a5ux^}�Æè1_b\ÆZ��{�9X¤z{��Z9ªuxw{Zr|��9\=~¦�9¡y��Zrú���~�a¤è�~��9�y\`ZrÁ{��d ^`��ux~�^=®1Z}^����G\`_ba5ue�t�y_A_b\e�â��~�´9Z��}\`~�u`Z]\=~�_ba[\`Z}�¦��u`~�|bZux_'cQøÇ��ú��j©¡5�«|A~�\=ux�{Z2_9��ú�� ¡y��_���´?~�ay�v�9����¡1�9��´b�â�y_�_9\oz1��u`wy^o�G\`_bX ø ux_µú���9^���Z}���$�9^�uxwy_b^=Z��G\=_bX�ú��rux_�ú�g��Þ�_��9����_��×u`wyZ��9a1�9����^=~�^r_9�Æ^=�y��wæ~�a5u`\`~��]��uxZ^=u`\`�y�4ux�y\=Z]^}ª�~�a{�]���{�y~�ay�[ayZ}^=uxZ}�¸�]_9X2¡y~�ay��ux~�_bay^�_9�Y¡1����´5�â�y_A_b\©��ay�¨�G\=_ba5u=�t�{_�_b\�]_9ay�y~�u`~�_ba{^]ª¬��X¤_9\`ZSz¬_���Z]\=�G�{��^=�AX2¡U_b��~��SX��9��w{~�ayZ}\=�¯��~�����¡UZH~�a5ux\`_A�y�y�}Z]�æ~�a¿�Z}�}ux~�_ba2Á��ÂÃyª�_bayZ¥uxw1��u���~����?^`~��{Z]^=u`Z]z7�����bZ}¡y\x�9~��£X��9a{~�zy�y����ux~�_bay^°^=�y��wF�9^�uxwy_b^=Z�y^=Z]�¾~�a,u`wyZv�yZ]\=~�|���u`~�_ba�_9�HcÀÁ{� � f5j�� ã��{u2®1\=^=uF��Z4u2�y^7��_�_9´��LuF�9a�Z4�{�9X[zy��Z~������y^eux\`��ux~�ay�HzU_b^`^=~�¡y��Z��9zyzy��~��]��ux~�_bay^�_9��uxw{Z��G\=_ba5u=�â�y_�_9\��]_ba{�y~�u`~�_ba/��m�g�m�g� z��j�0w¹�O¨k§����Ow¹¥ßô1°3­?yç��­[¦ ®> [§ ��§.­[¥�®��j�è§ ñ  [§dõ

¥����· _bay^=~��yZ}\ru`wyZS�]Z]a5u`�y\=�5�t_9���«�{Z]¡1��u`ZH_baµuxwyZS\`Z}�¦��u`~�_ba¯¡UZ}ut��Z]Z}aæ^`X[_b´?~�ay�¨cGø�j�9ay�'���{ay�¯�g�9ay�}Z]\¤cÀù2j7cJ¿�zy~�\=u`Z]^�Z}u��9�À��d�÷b÷9Á{ª�zyz/� � ÷{d$��Ábi � j��2½©�}�]_b\=�y~�a{�¯u`_X¤�9a5�bª¬u`wyZHu`_b¡1�9�}�]_�~�ay�y�y^eux\=�¨w1�9^©X��9ay�9�bZ]�µux_��G_9\`Z]^eu��9������a?u`~�^=X¤_b´?~�ay�¤��Z]���~�^=�¦��u`~�_ba[¡5�v�9\=�b�y~�ay�Huxwy��u�u`wyZP_b¡y^=Z]\=|9Z]�µ�]_b\=\`Z}�¦��u`~�_ba¤¡¬Z4ut��Z}Z]aµ^`X[_b´?~�ay�7��ay����{ay���g��ay�]Z}\o�]_b�{���¸¡UZ2Z4��zy���9~�a{Z]�¸¡5�¨^`_bX[Z2^`_9\=uo_��£�g�9\=�]~�a{_b�bZ]a{~��H�9Z]ay_9ut�AzUZc�Òoj�uxwy��u�~�a5|b_9��|bZ}^ ~�a?¡¬_b\=a¯�]\x�g|A~�a{�F�G_9\�ay~��}_9ux~�ayZ9�Þ�wyZ���X¤_b�{a?u�_���u���\�cÀúojP�yZ]zU_b^`~�uxZ}��~�aM�¯z¬Z}\`^=_ba/ûý^H���{ay�b^�~�^S�v|9��\`~¦��¡y��Z

uxwy��u�z{\`_bX[~�^=Z]^£ux_HX[Z]Z4u�u`wyZ©�}_bay�y~�ux~�_9ay^£��~�^euxZ}��~�a ²oZ}®yay~�u`~�_ba¤Á{�ÂÁ{� Á�ªAu`wA�{^�®yue�ux~�ay�ouxw{Z�^=ux\=�y�}u`�y\`Z�_9��è�~��b�y\`Z�Á{� ìA��Þ�_SX[Z]Z4u£�]_ba{�y~�u`~�_ba¯cG~Qj�ª9��Z�X2�{^=u¥�9^=^`�yX[Zuxwy��u�^`X[_b´?~�ay�F�}~��5�9\=Z}u=uxZ]^�w1��^ra{_¤Z4­�Z}�}ur_ba¯uxwyZ�zy\=_��y�{�}ux~�_ba¯_9�$���yay�F�]�9ay�]Z}\Z4���}Z]z{u��9^oX[Z]�{~¦��u`Z]�«u`wy\`_b�{�bw¸u��9\©�yZ]zU_b^`~�ux^}�SÞ�_vX[Z]Z}uo�]_bay�{~�ux~�_bay^2cD~�~¦jr��ay�cD~�~�~¦j4ª±��Z7X2�{^=uS�9^=^`�yX[Z2uxw1�Lugª/Z4|bZ]a�~��¥���bZ}ay_9ut�AzUZ[~�^P�9�b�9\x�g|���ux~�ay��uxw{Z7zy\`_���y�y�4ux~�_baæ_9�¥���ya{���g�9a{�]Z]\}ª°~�u©ayZ}|9Z]\=u`wyZ]��Z]^=^�w1��^Pay_vZ}­¬Z]�4u�_ba¸uxwyZF��X¤_b�{a?u©_9�u���\H~�a'uxwyZ����yay�9^HZ4���}Z]z{uF~�ay�y~�\=Z]�4ux���ÍcQuxwy\=_b�y�bwM�]~��b�9\`Z4u`uxZ¤^`X[_b´?~�ay�5j4�â��~�´9Z����~�^=Z9ª1��Z�X2�{^=u �9^`^=�yX[ZSu`w1��uray_[_9uxwyZ}\��D�9�}u`_b\�uxw1��u ��­¬Z]�4ux^�u��9\��yZ}z¬_9^`~�urw1��^

Page 28: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d�ÁbiyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê

��c �:/5j �¤cÀù !*d�� �:/5j�P\`_9�yz¨Þ£�Az¬Z ��\=_b�yz¨¿�~�ä}Z ��_9� · �9ay�}Z]\ · �9^=Z]^c�� _9��´�_bzy�y����ux~�_ba±j ~�aâ�P\`_b�{z

ø !×i�� ú !×i ér_bay^=X¤_b´�Z]\=^]ªUé _Fu��9\ Ãs��� ì d�iø !*d�� ú !×i ¿AX¤_b´�Z]\=^]ªUé _7u���\ � � ì ÷biø !×i�� ú !*d ér_bay^=X¤_b´�Z]\=^]ª±Þ���\ � � ì ìø !*d�� ú !*d ¿AX¤_b´�Z]\=^]ª±Þ���\ Ãs��� ì f5ì

Þ$�9¡y��ZSÁ{��d9ÿ�9a5�æ~�a��1�yZ}ay�]Z[_ba'^`X[_b´?~�ay�y�Fè$~�a1�9�����9ª��]_9ay�y~�u`~�_baÄ�¤c ��/5j�à�iv_9��Þ�wyZ]_9\`Z]XÁ{�ÂÁ{�ýÃ7\`Z]¢?�y~�\`Z}^�uxw1�Lu�wy~��9wv��Z}|9Z]��^�_9�°u���\�~�a�u`wyZP���{ay�b^�¡UZ©uxw{Zo\`Z}^`�y��u�ay_9u�_9ay���_9�U�]~��5�9\=Z}u`u`Z�^=X¤_b´?~�ay�o¡y�{u¥����^`_©_9�U_9uxwyZ}\��D�9�4ux_b\=^�cDZ��Â�y��ª5Z4��zU_b^`�{\`Z�ux_PZ}a5|�~�\`_baA�X[Z]a5u��9�±zU_b�����{u���a?u`^�jÆ�9a{�¤uxwy��u£u��9\£X¤���F¡UZ �9¡y^`Z}a5u�~�a¤^=_bX¤Zr^`X[_b´�Z]\`^©cD_���~�ay�zUZ]\`wy�9zy^�ux_r�9aSZ4�Aux\=Z]X[Z]���oZ}¶[�]~�Z]a5u$ux�9\e�â\`Zt³=Z]�}u`~�ay��X[Z]��w1��ay~�^=X�j��Æ¿{�Lux~�^e�D�9�}u`~�_ba_9��uxwy~�^��¦�9^eu��]_ba{�y~�u`~�_bav�]�9a¨¡UZPuxZ}^=uxZ}�«~�a�uxwyZ��1��ux�{�

Þ$_��yZ]X[_bay^eux\`��uxZ¯wy_�����Z¨�]�9a+�9^=^`Z}^`^¤u`wyZµ�yZ]�b\=Z]Z¯ux_'��wy~���w¼�]~��5�9\`Z4u`u`Z^`X[_b´?~�a{�H~�a{�]\`Z]�9^`Z}^�cD_9\��yZ}�]\=Zg�9^=Z]^�j����yay���â�g�9ay�}Z]\¥\`~�^`´±ª���Z©��~����±�9^`^=�yX[ZP�Hw5�5�zU_9uxwyZ4ux~��g�9��^=u`�y�{�«~�a¨��w{~���w¸uxw{Z�uxw{\`Z]Z�|9��\`~¦��¡y��Z}^rø��Pùz�Pú��Z}\`Z2X[Zg��^`�y\=Z]�^`~�X2�y��u��9ayZ}_b�y^=���2_9a����¦�9\=�bZ9ª?\x��ay�y_bX[���H^`Z]��Z]�4uxZ]��^x�9X[zy��Z�_9�¬uxwyZrzU_bzy�y����ux~�_ba/�Þ�_µ^=~�X[zy��~��Q�vuxwyZHZ4��z¬_9^`~�u`~�_ba�ª���Z7��~������G�y\=u`wyZ]\��9^=^`�yX[Z2uxwy��u��9����u`wy\`Z}ZF|��9\=~���9¡y��Z]^r�9\=ZS¡{~�a1�9\e�bªyux�9´?~�ay�[_ba¯ux\`�{Z�cOd;j�_b\��D�9��^=Z¤cÀi5j�|��9���{Z]^]�¥½w?�AzU_9uxw{Z}ux~��g�9��1��ux��^=Z}u¥�G\`_bXÚ�S^=ux�{�{�[_ba¤u`wyZ \=Z]����ux~�_bay^¥�9X[_bay�Pu��9\}ª��]�9ay�}Z]\]ª��9ay�¤�]~��5�9\`Z4u`u`Z^`X[_b´?~�a{�[~�^�zy\`Z}^`Z}a?u`Z]�¸~�avÞ���¡y��ZSÁ{��db�sâuo^=wy_���^ouxw1�Luo÷5ì���_��Y^`X[_b´9Z}\`^P��ay��ì���_9�£ay_9ay^`X[_b´9Z}\`^owy��|9Z2�yZ4|bZ]��_bzUZ]�wy~��bw¤��Z}|9Z]��^£_9�¬u��9\£~�a[u`wyZ]~�\¥���yay�b^}�Y§¨_9\`Z]_�|9Z]\]ª{f{d��*_��/^`�{¡A³=Z]�4ux^���~�uxwFu��9\£�yZ��zU_b^`~�ux^¥w1�g|bZo�yZ4|bZ}��_bzUZ]�����yay���g�9ay�}Z]\}ª{�]_9X¤z1��\`Z]�[ux_�_bay���¯d]Ã�� �9X[_bay��uxwy_b^=Z��~�u`w+ay_,ux�9\��yZ}z¬_9^`~�u`^]��è�~�a1�9�����bª¥��~�u`wy~�a+Z]�9��wÝ_9�Pu`wyZ]^=Z¸ut��_��b\=_b�yzy^æcGu���\�9ay�¼ay_��Ju��9\xj4ª£^`X[_b´�Z]\`^[^`w{_;����XH�y��wãwy~��bwyZ}\7z¬Z}\`�]Z}a5u��9�bZµ_9� �g�9a{�]Z]\Fuxwy�9aay_ba{^`X[_b´9Z}\`^]�

Þ�wyZ]^=Z«\=Z]^`�{��ux^F^=Z]Z]X u`_�z{\`_�|bZ¯uxw1�Lu¤^=X¤_9´A~�ay��~�^F��X���³=_b\2�]_9a?u`\`~�¡y�{ux_9\ux_«���yay�«�g�9ay�}Z]\}�ij _���Z4|bZ}\]ª£uxwyZ�u`_b¡1�9�}�]_æ~�ay�y�y^eux\=��X[~��bw5uH�9\`�b�{Z¤uxwy��u2uxw{Zu���¡y��ZruxZ]����^��H�y~�­¬Z]\=Z]a5u�^eux_b\e�5»+uxw1�Lu�^`X[_b´?~�a{�7�9�}u`�1�9�����[�yZ]�}\`Z]�9^`Z}^�_bayZ9ûý^�\`~�^`´

Page 29: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ ÌAκÄ�Ù�Ö7É$ËPÙ�Ñ�Ñ°ÓOÖHÔ"Ä�Ù�֠إÙ[Ï�Ö©ÒoÓOÖHÔÞí�ÓtÇPÐ d�Á�d_9�¬���ya{�S�]�9ay�}Z]\]�YÞ�w{Z]~�\¥�9\=�b�yX[Z]a5u¥�b_AZ]^£�9^Y�G_b����_���^]�$sâ���b_9�¤�yZ}�]~��yZru`_�^`X[_b´9Z�ªuxw{Z]aF�b_b�y\£��w1��ay�]Z}^�_9�¬¡y�y~����y~�ay�P�yz7u���\Y�yZ]zU_b^`~�ux^¥�9\=Z�÷5ì���ª?�]_bX[z1�9\=Z]�7ux_�ì��~����b_b�M�yZ]�}~��yZ[ay_9uSux_¨^`X[_b´9Z��vstaM_b\=�yZ]\�ux_¨Z}|��9���y��uxZFuxwyZ[Z}­¬Z]�4u2_9��u��9\S�yZ��zU_b^`~�ux^]ª���Z���_�_9´P^`Z]zy�9\x��u`Z]���2��u$ut��_P�9\`_b�yz{^]ª5^`X[_b´�Z]\`^Æ�9ay�7ay_ba{^`X[_b´9Z}\`^]��Þ�wyZu���¡y��Z7^=wy_���^Su`w1��u�u��9\P�{Z]zU_b^`~�ux^�w1�g|bZ¤�µz{\`_9u`Z]�}u`~�|9Z[Z4­�Z}�}u�~�a�¡U_9uxw'�b\`_9�yzy^]ÿ~�aæ^=X¤_9´9Z]\=^]ª°u���\P�yZ]zU_b^`~�ux^P��_;��Z]\P�]�9ay�}Z]\P\x�LuxZ]^P�G\=_bX�÷9i�� u`_¯f5ì���Û�~�aæay_baA�^`X[_b´�Z]\`^}ª¬u`wyZ}�µ��_���Z]\r�g�9a{�]Z]\ \x��u`Z]^r�G\`_bX d�i���ux_�ì����rÞ�w?�y^}ª�\=Z]�5�9\=�y��Z}^`^r_9���wyZ4uxwyZ}\Psrwy��|9ZF��a1��u`�y\x�9�$�]\`��|A~�ay���G_b\oa{~��]_�ux~�a{Z9ªUsr^=wy_b�y���桬ZH^`Z]Z}´?~�ay�vuxw{Zzy\=_9uxZ}�}ux~�|bZPZ4­�Z}�}ur_9��u��9\��yZ]zU_b^`~�ux^�~�avXH�����yay�b^}ªy�9ay�¯^=X¤_b´?~�ay�7_9­¬Z]\`^��H|9Z]\e�Z}­¬Z]�4ux~�|bZSX¤Z]�9ay^�_9���9�]¢?�y~�\`~�ay�7uxwy_b^=Z��yZ]zU_b^`~�ux^}�

Þ$_¯^=Z}u`u`��Z2u`wyZ7�y~�^=zy�{u`Z7¡¬Z4ut��Z}Z]a�u`wyZ2ut��_µ~�a5uxZ]\=zy\`Z4u���u`~�_ba{^]ª���ZFay_��"�9z��zy���«uxw{Z2�G\`_ba5u=�â�y_A_b\ �G_b\=X2�y���æcDZ}¢A�y��ux~�_baÍcÀÁ{� � ÷5j=jrux_�u`wyZ7�1��ux�¯~�aæÞ���¡y��Z7Á���db�å�Z2��~�^`w�ux_v�]�9���}�y�¦�LuxZHu`wyZFzy\`_b¡y�9¡y~���~�ut�¯uxwy��uS��\x�9a{�y_bX[���¨^`Z]��Z]�4uxZ]�'zUZ]\`^=_ba��~����Y�yZ}|9Z]��_bzM�]�9ay�]Z}\H�ya{�yZ]\�Zg�9��w,_���uxw{Z[�G_b����_���~�a{�¯ut��_¸���}ux~�_bay^}ÿH^`X[_b´?~�a{�cD^=Z}u=ux~�a{�Fø !d�j�_b\�ay_9u�^`X[_b´?~�a{�«cG^`Z4u`ux~�ay�7ø !×i5j4�¿��y¡y^eux~�ux�{u`~�ay�SuxwyZ �9zyzy\=_bzy\=~¦��u`Zr|��9���yZ]^¥_9���¤c4/^� j�����c-�.� �:/5j4ªA�9a{�¸�¤c j4ª��Z�w1�g|9Z

�¤cÀù !*d��<êsë{cQø !*d;j=j"! À i5ì{ckÀ�d�i 6 À ì�i�� ÀÂ÷bi 6 À ì9ibj�·ÀÂ÷5ìyc'À ibì 6 À ì�i � À fbì 6 À ì9i5j! À i5ì 6 À ì9i!� ÀÂ÷5ì 6 ÀýÃ?ì(! ÀýÃ?ì � ì���¤cÀù !*d��<êsë{cQø !×i5j=j"! À ÷5ì{ckÀ�d�i 6 À ì�i�� ÀÂ÷bi 6 À ì9ibj�·ÀÂi5ìyc'À ibì 6 À ì�i � À fbì 6 À ì9i5j! À ÷5ì 6 À ì9i!� ÀÂi5ì 6 ÀýÃ?ì(! ÀýÃ5÷o�bì�ÀºcÀÁ{�ÂÁbi5j

Þ�w?�y^]ªA�]_9a?u`\x�9\e�7ux_�Z4��zUZ]�4u���u`~�_ba/ª5u`wyZ��1��u��Pzy\=_�|bZr^`X[_b´?~�ay�Pux_�¡UZ�^`_bX[Z}��w1�Lu¡UZ]ayZ4®1�]~��9�/ux_F_bayZ�ûÂ^�wyZ]�9��u`w/�

Þ�wyZ��y��u��r~�aSÞ��9¡{��Z¥Á{��d��9\=Z¥_9¡?|A~�_b�y^`�����yay\=Zg�9��~�^=u`~��£�9a{����Z]\`Z��yZ]��~�¡UZ]\`��uxZ}����]\`���QuxZ}�¨^=_���^�u`_F^`�yzyzU_b\euru`wyZ��bZ]a{_9ut�Az¬Z�uxwyZ}_b\=�9�z³ _���Z4|bZ}\]ª±uxwyZ�zy�y\=z¬_9^`Z�_9�uxw{~�^2Z��{Z}\`�}~�^`Zv���9^2ux_æ�{Z]X[_bay^=u`\x��u`Z�wy_���\`Zg��^`_ba1��¡y��Z�¢?�1����~�ux��ux~�|bZ��9^`^=�yX¤zA�ux~�_bay^ �9¡U_b�{u©u`wyZ���_b\=´A~�ay�b^o_��YX¤Z}��w1�9ay~�^`X[^]ª¬�]_b�yz{��Z]�¨��~�uxw¨ay_bayZ���z¬Z}\`~�X[Z]aA�u����1�1��u���ªA�]�9a[zy\`_A�y�y�}Z zy\=Z]�}~�^`Zr¢?�1�9a5ux~�u���u`~�|9Z��9^`^=Z]^`^=X¤Z}a5ux^�_9���g�9�{^x�9�±Z}­¬Z]�4ux^}�sta,\=Zg�9��~�ut�bª���Z���_9�y���,Z���z¬Z}�}u7_b¡{^`Z]\e|���ux~�_ba1�9�£^=u`�y�y~�Z]^H~�a5|b_b��|�~�ay�µX¤Z}�y~¦��u`~�ay�|��9\`~��9¡y��Z]^ u`_v\`Z4�G�{uxZHu`wyZ7�bZ}ay_9ut�Az¬Z2u`wyZ]_9\=�¸¡5�«^`wy_���~�a{�yª��G_b\oZ��{�9X¤z{��Z9ªUuxw1�Luuxw{Z2X¤Z}�y~¦�Lux~�a{���]_9ay^`Z}¢A�{Z]ay�}Z]^H_��£^`X[_b´?~�ay��cG^`�y��w��9^©ux�9\o�yZ}z¬_9^`~�u`^�jru`Z]ay�¸u`_~�a{�]\`Z]�9^`Z�ªYay_9uH�yZ]�}\`Zg��^`Z9ªYu`wyZ�\`~�^`´�_9���]�9ay�}Z]\7~�a�^`X[_b´9Z}\`^7��ay��ay_bay^=X¤_9´9Z]\=^

Page 30: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d�Á � ÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê�9��~�´9Z���Þ�wyZ�Z}^=u`~�X¤�9ay�'_9�ScÀÁ�� � ÷bjP�}_b�y����uxw{Z]a,¡UZ¤�{^`Z]���G_b\�¢?�1��a?u`~��Q�A~�a{�¨uxw{Z�g���y^x�9�°Z4­�Z}�}u _9�$^`X[_b´?~�ay�7_baµ�g��ay�]Z}\]�

í¯î#" $ ê p �]q õ �gõ ��óFn ð�ñ�ò mPl��[m ñ�òYö ó ñÞ�wy~�^P^`Z]�4ux~�_baæZ]^eu��9¡y��~�^=wyZ]^���^=Z}uS_9�¥~�a{�GZ}\`Z}ay�]Z7\=�y��Z}^o¡5�¸��wy~���w�zy\=_b¡1�9¡{~���~�^=u`~��^`Z}a5uxZ]a{�]Z]^©~�a5|b_b��|A~�ay�F~�a5u`Z]\=|9Z]a5ux~�_bay^ �9ay�¨_b¡y^`Z}\=|���u`~�_bay^��]�9a«¡UZPux\x��ay^=�G_b\=X¤Z}�~�a5u`_�_9u`wyZ]\r^`�y��wæ^=Z]a5uxZ}ay�]Z}^]ª/u`wA�{^ozy\`_�|A~��{~�ay�¤�¤^e��a5ux�9�}u`~��2X[Z}u`wy_��µ_9�Y�{Z]\`~�|5�~�a{�'cD_b\©|9Z]\=~��Q�A~�ay�5j �]���9~�X[^o�9¡U_b�{u�~�a5uxZ]\e|bZ}a?u`~�_ba{^]�7W£�9��w�~�a��GZ]\`Z}ay�]ZF\=�y��ZH��~����\`Z}^`zUZ]�4uru`wyZ�~�a5uxZ}\`zy\=Z}ux��ux~�_bav_9��uxwyZ(êsë�c#%Ej�_9z¬Z}\x��u`_b\��9^r�9a¯~�a5u`Z]\=|9Z]a5ux~�_bavuxw1�LuX[_��y~�®1Z]^o��^`Z]��Z]�4u�^=Z}u�_9�¥�G�yay�4ux~�_bay^P~�a¸u`wyZ7�yay�{Z]\`���A~�ay�vX[_��{Z]�J�SÞ�wyZF^`Z4u�_9�~�a��GZ]\`Z}ay�]Z�\`�y��Z]^Hu`w1��u7Z}X¤Z}\`�bZ��G\=_bX uxwy~�^H~�a5u`Z]\`z{\`Z}ux��ux~�_baM��~����¥¡¬Z��]�9����Z]�Èê�ë@`>9C�@4çACEç?:4�åMZ¨��~���� �9^=^`�yX[Z«uxwy��u���Zæ�9\`Z«�b~�|9Z]a+uxw{Z¸^=u`\`�y�4ux�y\=Zæ_9���M�g�9�{^x�9� �y~¦�L�

�b\`�9X¹±~�a���wy~���wv^=_bX[Z©_9�°u`wyZ©ay_A�yZ}^��9\`Z©_b¡y^`Z}\=|��9¡y��Z ��wy~���Zr_9uxw{Z]\`^�\`Z]X¤�9~�a�yay_9¡y^`Z}\=|bZ}�/�ײP�{\¤_b¡?³=Z]�}u`~�|9Z«��~�����¡¬Zµux_��D�9�]~���~�u���u`Z�uxwyZµ^=�Aa5u��9�4ux~��«�yZ}\`~�|��L�ux~�_ba,_9���]�9�y^`�9��Z}­¬Z]�4uFZ4��zy\=Z]^`^=~�_ba{^[_���uxwyZ��G_9\`XÆ��c-�.��� j4ªÆ��wyZ}\`Z�ø �9a{�¾ù^=ux�9ay�«�G_9\o�9a5�µ^`�y¡{^`Z}u`^P_9�£_b¡y^=Z]\e|bZ]�æ|��9\=~¦�9¡y��Z]^}�jã¥�Ãà=�{Z]\`~�|���ux~�_ba�á7��ZHX¤Z]�9a^=u`Z]z5��~�^=Z©\=Z]�y�y�4ux~�_ba¤_��¬u`wyZ Z���zy\`Z}^`^`~�_ba¸�¤c������ jÆu`_H��a¤Z}¢A�{~�|��9��Z}a5u�Z���zy\`Z}^`^`~�_ba~�a5|9_b��|A~�a{�S^eu��9a{�1�9\`�[zy\=_b¡1�9¡y~���~�ux~�Z}^�_9��_b¡y^=Z]\=|9Z]�¤¢A�y�9a5ux~�u`~�Z}^]��åÜwyZ}ayZ}|9Z]\�^`�y��w\`Z}�y�y�4ux~�_9aã~�^7�GZ]�9^`~�¡y��Z�ª�uxw{Z¨�]�9�y^x���rZ4­�Z}�}u�_9�©ø _baãù ~�^F~��yZ]a5ux~�®±�9¡y��Z�cD^`Z}Z²oZ}®yay~�u`~�_ba¯Á{� � �ýÃ?j4�

�m��Îj�3� & �>§.¨k°�w¹°k­[�����(' ¥�®��®�°'¥�­O­[¥�®��®�°'¥�­�°Z}u�øÇ�Sù����9ay�Mú ¡¬Z[�9\=¡y~�u`\x�9\e�«�y~�^G³=_b~�a5u�^`Z}u`^�_���ay_A�yZ}^S~�a'�¯�]�9�y^`�9��²r½1�±F��åMZr�yZ}ay_9uxZ ¡5�ѱ Ï uxwyZr�b\`�9zyw�_9¡{u��9~�ayZ]�[¡5�[�yZ]��Z}ux~�ay�P�G\`_bX ±��9���1�9\=\`_���^zU_b~�a5ux~�ay�oux_�ay_A�yZ}^�~�aFø,���°~�´�Z}��~�^=Z9ª?��Zr�yZ}ay_9uxZ�¡5�ݱOÏ uxwyZ��b\`�9zywF_b¡{ux�9~�ayZ}�¡5���yZ}��Z4ux~�a{�H�G\`_bX ±�9���¬�9\`\=_���^�Z]X[Z]\`�9~�ay�H�G\=_bXÚay_A�yZ]^�~�a�ø��YÞ�_7\`Z]z{\`Z]^=Z]a5uuxw{Z¤�yZ}��Z4ux~�_9a�_9��¡¬_9u`wM~�ay�]_bX[~�a{�¯�9ay�,_b��ux�b_b~�ay�µ�9\`\=_���^]ª$��Z��{^`Z¤u`wyZ�a{_9u��L�ux~�_ba ± Ï*) cD^=Z]Z¯è�~��b�y\=Z�Á{�Âh¨�G_b\��9a,~������y^=u`\x��u`~�_ba±j��¯è�~�ay�9�����bª�uxwyZ�Z���zy\`Z}^`^`~�_ba�¤c������ �:/bj +! ��c-�^��/^��� j�¯;�¤c4/^��� j�^=u���ay�y^��G_b\�u`wyZ�zy\`_b¡y�9¡y~���~�ut�[_9�$ù !����9~�|bZ}auxwy��u ø ~�^rwyZ]���«�]_9ay^=ux�9a5u���u �9ay�¨u`w1��u7cD�yay�{Z]\ u`wy~�^r�]_9ay�y~�u`~�_ba1jrú ! /�~�^_b¡y^=Z]\e|bZ]���

Page 31: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ-,AÎÍÇ Ä�Ç©Ñ¥ÄPÏ$ÑrÏ�Ð�Ù�Ø�ÓOÖ7É$Ê¥ËmÊ�Ê¥Ö7É�Ó4Ù�Ö d�Á9Á�m��Îj��� «s­[ª�§.�>§.­?~V§/.�xO¨'§.¡Þ�wyZ©�G_9����_���~�ay�Su`wyZ]_9\`Z]X ^=ux��uxZ}^�uxw{Z uxw{\`Z]Z©¡1�9^`~��©~�a{�GZ]\=Z]ay�}Zo\`�y��Z]^�_9�°uxwyZ©zy\`_��zU_b^`Z}�¨�]�9���}�y���y^}��´Y\`_A_9�G^r�9\`ZPzy\=_�|�~��yZ]�µ~�a¬´�Zg�9\=�¥ced�÷b÷bì9�5j4���� ����b 2� ���âá��� "!10£&)+� 2(;��¡1êsëA#%'+�1�&)+�&)(I4¥�04Bm± 840¤BQ.10¯T9IQßx0�@4Bt0`T'>A@435@4CEIJ@ �9ß=>Q=1.�>L:�:�6�@4ID>9Bt0`TÐ6YIQBG.�>�@`>9ç?:]>9C¥R�6]TA04C¥>L:TA0�Í£K�0`T¨IQK¾cDÁ{� � j98o>9KUT¨C�04B���c2%Ej�:4BJ>9KUT�Ng6Lß[BQ.10?=Ußx698�>58]IQCEIQBD3¸T9I�:4BDß�ID8]ç�BDIJ6LK,IQK>ØT9ç1@�0`T�8]3FBG.{>9B¥R�6]TA04C�Vr@�6Lß�>9K13¤T9I�:�7g6LIQK1BY:4ç{84:�04BQ:26ONPàL>9ß�ID>58]C�0�:�øÇ�xùz��úX8�>9KUT3 6�0o.{>9à90�BQ.10£Ng6LCQC�6b6YIQK��[ß�çAC�0�:]V4�çAC�065+c¿Û4K{:�04ß�BÀIJ6LK�7�TA04C�04BDIJ6LK'6ON26984:�04ß�àL>9BDIJ6LK{:`j�ÿ

�¤c������ �:/��98ojz! ��c-�.�4� �98ojÚI N cDù;:<:HúÝ� øÇ�=3�j#> ?�À cÀÁ{�ÂÁ{d;j4�çAC�0A@Üc ú @4BDIJ6LK�7°6984:�04ß�àL>9BÀIJ6LK�0=<�@�.{>9K��A0`j¥ÿ

�¤c������ ���/��98Pj�!���c-�.��� �:/���8Pj I NÚcÀù;:<:2úÝ� øÇ�=3�j#> ?CB À cÀÁ{�ÂÁ � j4�çAC�0EDÝc¿Û4K{:�04ß�BÀIJ6LK�7�TA04C�04BDIJ6LK'6ON�>A@4BDIJ6LK{:`j�ÿ

�¤c������ ���/��98Pj�!Ì�¤c������ �98Pj"I N cDùF:G:�úÝ� ø��H3�j1> ?JI BLKNM*O � cÀÁ{�ÂÁbÁ5j6�.104ßx0SúHc�3�jFI�:2BG.10H:�04BP6ON�ú,ØtK�6]TA0�:7BQ.{>9B >9ß�07K�6LB©>9K�@�0�:4Bt6Lßx:�6ON[>9K133 ØtK�6]TA0�IQK ± Ï VW£�9��wF_9�1uxw{Z]^`Z�~�a{�GZ]\=Z]ay�}Z�\`�{��Z]^��G_b����_���^$�G\`_9X�u`wyZ�¡1��^`~���~�a5uxZ}\`zy\=Z}u��Lux~�_9aH_9�±uxw{Zà=w1�Lu�á � _bzUZ]\x�Lux_b\©�9^P��\`Z}zy�¦���]Z]X[Z]a5uP_��¥u`wyZ2�]�9�y^x���ÆX¤Z}��w1�9ay~�^`X uxw1�LuP�]_baA�ayZ}�}ux^$ø�ux_o~�ux^$zy\`Z]�9�}u`~�_ba�z1�9\`Z}a5ux^Y¡5�2� ayZ}�¼X[Z]��w1�9a{~�^`Xþø ! ~�a5u`\`_A�y�y�]Z}�¡5�FuxwyZr~�a5uxZ}\=|9Z]ay~�ay�H�G_b\=�]Z9��Þ�wyZ©\=Z]^`�{��u�~�^��S^=�y¡yX[_��yZ}�¬��wy�9\x�9�4uxZ}\`~�ä}Z]��¡5�Fuxw{Z^`�{¡y�b\x��zyw�± Ï cDay�9X¤Z}�¢à=X���ay~�zy�{�¦��u`Z]�v�b\`�9zyw�áF~�a¨¿�z{~�\=u`Z]^�Z4u©���J�/d�÷9÷bÁ5j4�

êr�{��Zæd¯\`Z]��¶[\`X[^�êb�â^`Z}z1�9\x�Lux~�_9a��9^[�¸|9����~���uxZ]^euF�G_b\2�]_9ay�y~�u`~�_bay�9��~�ay�yZ4�zUZ]ay�yZ}ay�]Z�~�aHuxw{Z��y~�^eux\=~�¡y�{u`~�_ba�\`Z}^`�y��ux~�a{� �G\`_9X�u`wyZ�~�a?u`Z]\e|bZ]a5u`~�_ba�êsë�cGø ! j4ªwyZ}ay�]Z�uxw{Z��b\x�9z{w�± Ï �ÆÞ�wy~�^Æ\`�y��Z��G_b����_���^��G\`_9Xu`wyZ��D�9�}u�uxw1��uÆ�yZ]��Z}ux~�ay�PZ]¢?�1�;�ux~�_bay^��G\=_bXÚuxwyZP^e��^euxZ}X��y_�Z}^�ay_9u�~�a5ux\`_A�y�y�}Z��9a5���yZ}z¬Z}ay�yZ}ay�]~�Z]^r�9X[_bay�2uxw{Z\`Z}X���~�ay~�ay�2�y~�^=ux�{\`¡1�9a{�]ZPuxZ}\`X[^HcG^`Z}Z�cDÁ{� � j`j4�

êr�{��Z � zy\=_�|�~��yZ]^©�¤�}_bay�y~�ux~�_9a¯�G_9\©�9a¨Z4�AuxZ}\`a1���$~�a?u`Z]\e|bZ]a5u`~�_ba�ê�ë�cJú�! /5jux_µw1�g|bZ¤uxwyZ[^x�9X[Z[Z}­¬Z]�}u�_baMù&�9^�uxw{Z¤z1��^`^`~�|bZ¤_b¡y^`Z}\=|���u`~�_ba,ú ! /��vÞ�wyZ

Page 32: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d�Á9Ã1ÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê�]_9ay�y~�u`~�_ba¤�9X[_b�ya5ux^£ux_ Þ�ø{aP3 ßP¡{��_A��´A~�ay�2�����1¡1�9��´b�â�y_�_9\�z1��u`wy^��G\=_bX úÜu`_ù�cD~�aâ± Ï j4ªy^=~�ay�}Z,± ÏQ) \=Z}ux�9~�ay^������£cD�9ay�¯_bay���yj�^=�y��w«zy��uxwy^}�êr�{��ZoÁ�zy\`_�|A~��yZ}^��]_9ay�y~�u`~�_ba{^¥�G_b\�~�a5ux\=_��{�y�]~�ay�vcG_b\��yZ}��Z4ux~�a{�?j¥��a�Z4�AuxZ}\`a1���~�a5u`Z]\=|9Z]a5ux~�_ba êsë{cÀúÆ! /5jF��~�uxw{_b�{u���­�Z}�}u`~�ay�Mu`wyZ¸zy\=_b¡1�9¡{~���~�ut��_9�Sù ! �¬�Þ�wyZo|��9��~��y~�ut�F_9�°uxwy~�^�\`�{��Z©^euxZ}X¤^}ª1�9�5�9~�a/ª?�G\`_bX ^`~�X2�y�¦�Lux~�a{�Su`wyZP~�a5uxZ}\=|9Z]a5ux~�_baêsë{cÀú !Ì/5j/¡5�ouxwyZ£�yZ}��Z4ux~�_9aP_9�y�����9Z}¢?�1��ux~�_bay^��]_9\`\`Z}^`zU_bay�y~�ay��ux_�uxw{Z£|��9\=~¦�9¡{��Z]^~�aMú�cDwyZ}ay�]Z[uxwyZF�b\`�9zyw ± Ï ) j���Þ�wyZ¤\`Zg��^`_ba��G_b\���~�X¤~�ux~�a{�¤uxw{Z¤�yZ}��Z4ux~�_9a�u`_ay_bay�9ay�]Z}^=u`_b\`^2_��!3á�tay_A�yZ]^S~�^Szy\=_;|A~��yZ]�'��~�uxw'uxwyZ¤zy\`_A_9�G^S_9��ê��y��Z}^¤d$��Á¨~�a´$Z]�9\`�¥cOd�÷b÷5ì��5j4�#���b��+�+�%[�IÊK���âá���� ú @`>9ç?:]>9C�0�¦o0�@4BSR�!Ö�¤c��s���À8À�À8���¶���� ���À8À�À8��� CT j�I�:�IDTA04K1BDIÙØÍ�>58]C�0æIQKº>ÍR�6]TA04CS@�.{>9ß=>A@4Bt04ß�ICBL0`Tã8]3Í>È�9ß=>Q=1.ܱ I NæBQ.104ßx0M0=<bI�:4BQ:�>�Í£K1IQBt0:�0FE]ç104K�@�0µ6ON¤BDß`>9K{:GNg6Lß�R[>9BDIJ6LK{:98H0`>A@�.¼@�6LKgNg6Lß�R7IQK��æBt6'6LK�0¨6ON¤BG.10�IQKgNg04ßx04K�@�0ß�ç�C�0�:SIQKÍ-/.10�6Lß�04RUD?VWVUVX5b8£BG.{>9B¥ßx0`T9ç±@�0�:YR¨IQK1Bt6�>F:4BJ>9KUT5>9ß`TvcOIDVD0]V�Z¦.{>9B=[�Ø N�ß�0�0`j=Ußx698�>58]IQCEIQBÀ3�0=<*=Ußx0�:�:4IJ6LK¸IQK1à96LCEà�IQK���6984:�04ß4à90`T¢E]çy>9K1BDIQBÀIJ0�:]VåÜwyZ4uxwyZ}\oê��y��Z}^�dQ�AÁ��9\=Z�^=�{¶[�]~�Z}a5u©�G_b\r�yZ]\=~�|A~�a{���9����~��yZ}a?u`~�®±��¡y��ZP�]�9�y^x���$Z4�¦��GZ]�4ux^Æ\=Z]X¤�9~�a{^Æ�9a2_9z¬Z}a7¢A�{Z]^=u`~�_ba���³ _���Z}|9Z]\]ª5u`wyZ�u���^`´S_9�1®1ay�y~�ay�P�©^=Z]¢?�yZ}ay�]Z_9�yu`\x�9a{^=�G_b\=X���u`~�_ba{^�cG~��{^`�y��w7Z��{~�^=u`^�j��G_9\$\=Z]�y�y�}~�ay�©�9a7�9\=¡y~�u`\x�9\e�P�g���y^x�9�AZ}­¬Z]�4uZ4��zy\=Z]^=^`~�_9aÝ�g�9aÝ¡UZ«^e��^euxZ}X���u`~�ä}Z]�Ü�9ay�ãZ4��Z]�}�{uxZ}�á¡5�ÍZ}¶[�]~�Z]a5u¯�9���b_b\`~�uxwyX[^c����9����Z]^¤�9ay�¤´$Zg��\`�Hdg÷b÷5ì�Ûõ´$Zg��\`�o��ay�Üêr_9¡y~�ay^µd�÷b÷bìbj4ª�u`_,¡¬Z«�y~�^=�]�y^=^`Z]�Ü~�a· w1�9z�uxZ]\¥Ãy�ƽ ^¥��Z©~������y^=u`\x��u`Z�~�a�¿�Z]�4ux~�_ba�Á��ÂÃy�ÂÁ{ª?^=�AX2¡U_b��~����yZ}\`~�|���u`~�_ba{^Y�y^`~�ay�uxw{ZHw1��u ay_9ux��ux~�_ba¨�9\`ZSX2�y��w¸X[_b\=Z��}_ba5|bZ]a{~�Z]a5uou`w1�9a«�9���bZ}¡y\x��~����yZ]\=~�|���u`~�_bay^uxwy��uP�9~�X ��uoZ}��~�X¤~�a1��ux~�ay�7�¦�LuxZ]a5u |9��\`~¦��¡y��Z}^r�G\=_bX ^=u���ay�1�9\=�¸zy\`_9¡1�9¡y~���~�ut�¯Z��?�zy\=Z]^`^=~�_ba{^2cD�9^�~�a¨¿�Z}�}ux~�_baµÁ{�ÂÁ{� � ª1Z]¢?�1��u`~�_ba�cDÁ{� � Ã?j`j���m��Îj�g� ���Ow Lè¥�¨3°¹~Ýö̧��w°�¯��®�°k¥�­ ¥�ªÚvÈ�0x[¡���¨KzA| §�~|®�¡\�^] ­z��j�0w¹�O¨k§å�Z���~����?ay_��¾�yZ]X[_bay^eux\x�LuxZ�w{_;�Íê��y��Z}^�d$��Á �]�9a7¡¬Z��y^`Z}�2ux_ �yZ]\=~�|bZ��9���5�]�9�y^x���Z}­¬Z]�4u©Z}^=u`~�X¤�9ay�y^�~�a¯u`wyZ�^=u`\`�y�4ux�y\=ZH_9��è�~��9�y\`Z�Á{� ì��£è�~��b�y\=Z�Á{� hF�y~�^`zy�����A^�uxw{Z^`�{¡y�b\x��zywy^�uxw1��u���~����¬¡¬Z�ayZ}Z]�yZ}�¨�G_b\�u`wyZ��yZ]\=~�|���ux~�_bay^�u`w1��u��G_b����_����� %'(*ùÞ '²t#���ɱÕ&��� Ç�¤c4/���� jÞ�wy~�^rux�9^`´v�]�9a«¡UZH���]�]_9X¤zy��~�^=wyZ]�µ~�aµ_bayZS^=u`Z]z/ªU^`~�ay�]Z,±"^x��u`~�^=®yZ]^�uxwyZ��9zyzy��~���g��¡y~���~�ut�¤�]_bay�{~�ux~�_bav�G_b\�ê��y��Z � �£Þ�w1��ur~�^]ªyø_:<:Húþ~�a¬±OÏ cD¡UZ]�g���y^`Z�uxwyZSz1��u`wø^� Ò c ù�� ú*~�^o¡{��_A��´9Z]�æ¡5�¨uxwyZ2�}_ba5|bZ}\`�b~�ay�¯�9\`\=_���^���uPùHj©�9ay�¸��Z�g��aµ��\=~�uxZ ��c�/^��� j�! ��c�/^� j�À cÀÁ{�ÂÁ9Ã?j

Page 33: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ-,AÎÍÇ Ä�Ç©Ñ¥ÄPÏ$ÑrÏ�Ð�Ù�Ø�ÓOÖ7É$Ê¥ËmÊ�Ê¥Ö7É�Ó4Ù�Ö d�Ábì

X Z Y X Z YX Z Y

X Z Y

U (Unobserved)

X Z Y

Z X

Z XZXZG G G

G = GG

è�~��9�y\`Z�Á{�Âh{ÿ¥¿��{¡y�b\x��zywy^�_9�± �y^`Z}�¨~�a¯u`wyZ��yZ]\=~�|���u`~�_bav_9�$�g���y^x�9�°Z4­�Z}�}u`^]�

� %'(*ù���²t#���ɱÕ&��� Ç�¤c������/9j³ Z}\`Z���Zr�]�9ayay_�u¥�9zyzy���7ê��y��Z � u`_PZ4����w1�9ay�9Z¬�/P��~�u`w�/S¡UZ]�]�9�y^`Z1±E) �]_9a?ux�9~�a{^� ¡y�9��´b�t�y_A_b\�z1��uxwS�G\`_bXú�ux_©ù¤ÿ$ú`� øa� Ò c ù¤��é©��u`�y\x�9�����9ª;��Z���_b�y�����~�´9Z�ux_�¡{��_A��´�uxwy~�^©z1��u`w¸¡5�«X[Zg��^`�y\=~�ay�F|��9\`~��9¡y��Z}^2cG^`�y��w���^ ø'j�uxw1��u©\`Z}^`~��{Z_ba«u`w1��u©z1��uxw��©Þ�wy~�^©~�a5|9_b��|9Z]^o�}_bay�y~�ux~�_9ay~�ay�¤�9ay�«^`�yX[X¤~�ay��_�|9Z]\P�9���°|9�����yZ}^_9��ø�ÿ

�¤c������/�jz!Ì# 7 �¤c���� �s�/bj3�¤c �4�/5j�À cÀÁ{�ÂÁ5ìbjåMZ«ay_��&wy��|9Z¸ux_,�{Zg�9�r��~�uxwÍut��_,u`Z]\`X[^�~�a5|b_b��|�~�ay� �/��¸��c-�.� ���/5j¤��ay��¤c ���/bj4�¼Þ�wyZ¯�¦��u=uxZ]\7�]�9a¼¡¬Z¯\`Z]�9�y~����,�}_bX[zy�{uxZ}�Í¡5�¼�9z{zy���A~�a{��ê��y��Z¯Áæ�G_9\�9�4ux~�_9aµ�{Z]��Z4ux~�_ba/ÿ

�¤c ���/bjz! ��c j!~��!cJúb:<:�ø�j1> B^� cÀÁ{�ÂÁbh5j^`~�ay�]Z¯ø ��ay�Ýú��9\`Zâê9�t^`Z}z1�9\`��uxZ}�ã~�a¢± ) ��cGsta5ux�y~�ux~�|9Z]���bª�X¤�9ay~�zy�y�¦�Lux~�a{��ú^`w{_b�y���¯w1�g|9Z�ay_FZ4­�Z}�}u _bavø�ªy¡UZ]�g���y^`ZHúº~�^��7�yZ]^=�]Z}ay�1�9a5uo_��$ø ~�a¬±F� j2Þ�_\`Z}�y�y�}ZSu`wyZP�G_b\`X[Z]\�u`Z]\`Xµª>�¤c���� �s�/9j4ª{��ZS�]_bay^=�y��urê��y��Z � ÿ

�¤c���� �s�/bj�!��¤c���� �:/bj!~��ÝcJúb:<:�ù¬� ø'j#> B � cÀÁ{�ÂÁo�bjay_9u`~�ay�µuxw1�LuHø ê9�t^=Z]z1�9\`��uxZ}^7ú��G\=_bX ù&~�a ±6) �«Þ�wy~�^7�9����_���^��{^Hux_«��\=~�u`ZcÀÁ�� Á5ì9j��9^

�¤c������/bj�! # 7 �¤c���� ��/5j3�¤c j�!ÃG 7 �¤c���� ��/5j�� cÀÁ{�ÂÁbf5j

Page 34: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d�ÁbhyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê��wy~���w�~�^���^=z¬Z}�]~¦���U�]�9^`Z _9�/u`wyZ©¡1����´5�â�y_A_b\¥�G_9\`X2�y���[cDZ]¢?�1�Lux~�_9aæcÀÁ���d�÷bj`j4��Þ�wyZ��Z}�b~�u`~�X[~�ä}~�ay�[�]_ba{�y~�u`~�_ba/ª�cÀúb:G:Sù¬� ø'j1> B ª$_9­¬Z]\=^P�bZ}u2�9ay_9u`wyZ]\��b\`�9zywy~��g�9�$uxZ]^eu�G_b\r�F^`Z}u ø u`_¤¡UZS^`�{¶[�]~�Z]a5u©�G_9\r�}_ba5ux\=_b��_9�Æ�}_ba{�G_b�ya{�y~�ay�«cD¡UZ}ut��Z]Z}a�ù���ay�úoj©uxwy��uS~�^PZ]¢?�y~�|��9��Z}a?u�u`_¯u`wyZ[~��bay_9\x�9¡y~���~�ut�¨�}_bay�y~�ux~�_baæ_9��ê�_b^=Z]a?¡1�9�yX ��ay�ê��y¡y~�a�cOd�÷bfbÁ5j��� %'(*ù���²t#���ɱÕ&��� Ç�¤c������ jå�\`~�u`~�ay� ��c-�.��� j�! #�$ ��c-�.� /���� jk��c�/^��� j�� cÀÁ{�ÂÁb÷5j��Z¨^=Z]Z¯uxw1��uFu`wyZvuxZ]\=X �¤c4/^��� j����9^[\`Z}�y�y�]Z}�+~�aácDÁ{� Á�Ã?jH¡y�{uFuxwy��u[ay_�\`�y��Z�g��aµ¡UZS�9zyzy��~�Z}��ux_7Z]��~�X[~�a1�LuxZ u`wyZPw1��u�^e�AX2¡¬_9�(�7�G\=_bX�u`wyZouxZ}\`X �¤c����</���� j4�³ _���Z}|9Z]\]ª±��Z��]�9a¯��Z]�9~�ux~�X��LuxZ]���[�9�y�¯uxwy~�^�^=�AX2¡U_b�/|A~¦�Fê��y��Z � ÿ

��c-�.� /���� j�! ��c-�.���/���� j�� cÀÁ{�ýÃ5i5j^`~�ay�]ZSuxwyZH�9zyzy��~��]�9¡y~���~�ut�¤�]_ba{�y~�u`~�_ba�cÀù;:G:�úÝ� ø'j#> ?CB wy_b���{^2cG^`Z]ZHè�~��9�y\`ZSÁ{�Âh5j4�å�Zæ�g�9a!ay_����yZ}��Z4uxZ¸uxw{Z��9�}u`~�_ba � �G\`_bX ��c-�.���/���� j��y^=~�ay�¼ê��y��Z¸Á{ª©^=~�ay�}Zù;:<:�ø � úºwy_b���y^�~�a�± Ï ) �¥Þ�w?�y^}ª1��ZSw1�g|bZ

�¤c����</���� jz!���c-�.���/bj�� cÀÁ{�ýÃyd;j��wy~���w«����^o�g�9���]�y����uxZ}�¨~�a�cDÁ{� Á9f5j4� ¿��y¡{^=ux~�ux�{u`~�ay�¸cÀÁ{�ÂÁbf5j�ª$cÀÁ{�ýÃyd;j�ª��9a{�¾cÀÁ{�ÂÁ9Ã?j¡1�9��´v~�a5ux_µcÀÁ�� Áb÷bj�®1a1�9�������A~�Z]���{^

�¤c������ j�!¶# $ ��c�/^� j>#87 9 ��c-�.� >= �:/5j3�¤c >= j�� cÀÁ{�ýà � j��wy~���w¨~�^�~��yZ}a?u`~��]�9��ux_7u`wyZP�G\`_ba5u=�â�y_A_b\��G_b\`XH�y�¦�H_���cDÁ{� � f5j4�� %'(*ùKá�²t#���ɱÕ&��� Ç�¤c����:/^�4� jå�ZSwy��|9Z �¤c����:/^�4� j�!��¤c����</���� j3�¤c4/^��� j�ÀÞ�wyZ[ut��_µuxZ]\=X¤^S_ba'uxwyZ[\]�ýw/�ý^]����Z}\`Z��{Z]\`~�|bZ}�,¡¬Z4�G_b\`Z[~�aãcÀÁ�� Á9Ã5j���ay�ÝcDÁ{�ÂÃ{d;j4ª�G\`_9X���wy~���wµ��Z�_9¡{u��9~�a

��c-�^�:/���� jÎ!��¤c������/9j3�¤c4/^� j!��¤c4/�� jC� 7 9 ��c-�.� = ��/5j3�¤c = j�À cÀÁ{�ýÃ5Á5j� %'(*ù�æ�²t#���ɱÕ&��� Ç�¤c �Z�.���/bjå�ZSwy��|9Z��c �������/5jç! �¤c���� ���/bj3�¤c �4�/?j! �¤c���� ��/5j3�¤c j�À cÀÁ{�ýÃbÃ?j

Page 35: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ-,AÎÍÇ Ä�Ç©Ñ¥ÄPÏ$ÑrÏ�Ð�Ù�Ø�ÓOÖ7É$Ê¥ËmÊ�Ê¥Ö7É�Ó4Ù�Ö d�Á��Þ�wyZo®1\=^=u�u`Z]\=X�_ba�u`wyZo\}�Âw/�ý^]�{~�^�_9¡{u��9~�ayZ]��¡?��êr�{��Z � cD��~��]Z}ay^`Z}�v¡?�¸±6) j���ay�uxw{ZS^=Z]�}_bay�¨u`Z]\=X�¡5�¯êr�{��Z�Á¨cÀ�9^�~�aMcDÁ{�ÂÁbh5j`j��é _�uxZ¤u`w1��u]ª�~�aM�9���Yu`wyZ��yZ}\`~�|9�Lux~�_9ay^]ª�uxwyZ¤�b\x�9z{wÕ±�wy�9^Hzy\=_�|�~��yZ]�M¡¬_9u`wMuxw{Z��~��]Z}ay^`Z7�G_b\S�9zyz{���A~�ay��u`wyZ[~�a{�GZ}\`Z]a{�]Z[\`�y��Z]^S�9ay��uxwyZ[�b�y~��1�9ay�}Z7�G_b\P��wy_A_b^`~�ay�uxw{ZS\=~��bw5u�\=�y��Zou`_¤��zyzy���9�

�m��Îj��Î v �0x[¡���¨ «s­[ª�§��>§.­?~V§ LS�c�Ox[�>�>¥ßy��®�§ zd�õ�è§��w°�w §.­�®�¡¿��yz{z¬_b^=Z2��Z7��~�^`w¸ux_v��Zg�9\=a«uxw{Z7�g�9�y^`�9�ÆZ4­�Z}�}u�_9�£ø _ba�ù���w{Z]a ��c-�.��� j ~�^ay_9u�~��yZ]a5u`~�®±�9¡{��Z2��ay�/ª°�G_b\ozy\`�9�}u`~��]�9��\=Zg�9^=_bay^�_9�¥�]_b^euS_b\oZ}u`wy~��}^]ª°��Z[�]�9ayay_�u�]_9a?u`\`_b��ø ¡5�¸\x��ay�y_bX[~�ä}Z]�¸Z��{zUZ]\=~�X[Z]a5ug��Þ�wyZ7¢?�yZ}^=u`~�_ba��9\=~�^=Z]^o_9�£��wyZ4uxwyZ}\�¤c������ j��]�9a¯¡¬ZP~��yZ]a5u`~�®1Z}�µ¡5�v\`�9ay�y_9X¤~�ä]~�a{�2�2^`�{\`\`_9�5��uxZo|��9\=~¦�9¡{��ZPú×u`w1��u�~�^Zg��^`~�Z}\�u`_7�]_ba5ux\=_b�¬u`w1�9a�ø,�£è±_9\�Z��y��X¤zy��Z9ª�~��/��Z��9\`Z ~�a5uxZ}\`Z}^=uxZ}�¯~�av�9^=^`Z}^`^`~�ay�uxw{Z¤Z4­�Z}�}u2_�����wy_b��Z]^euxZ]\=_b�£��Z4|bZ]��^�cQø'jo_baMw{Zg�9\euH�y~�^`Z]�9^`Z«cÀùHj4ªÆ�¯\`Z]�9^`_9a1�9¡y��ZZ4��zUZ]\=~�X[Z]a5uHu`_«�]_9ay�y�y�4uH��_b�y���M¡¬Z[ux_«�}_ba5ux\=_b��^=�y¡A³=Z}�}ux^}ûÆ�y~�Z}u�cJúoj�ª�\x�LuxwyZ}\uxwy�9avZ4��Z]\=�]~�^`~�a{�7�y~�\=Z]�}u��]_ba5ux\=_b�/_�|bZ}\���wy_b��Z]^=u`Z]\=_b�/��Z4|bZ}��^�~�a�^`�y¡?³=Z]�}u`^]û1¡y��_�_A�/�

è1_b\`X¤�9�����bª�uxw{~�^Æzy\=_b¡y��Z]X�9X[_b�ya5ux^�u`_©ux\`�9ay^e�G_b\`X[~�ay�õ��c-�.�4� j$~�a5ux_oZ��{z{\`Z]^O�^`~�_bay^ ~�a¨��w{~���w¸_ba{���µX¤Z}X2¡UZ]\`^r_9��ú*_9¡{u��9~�aµu`wyZ2w1�Lu©^e��XH¡¬_b�À�/år^`~�ay��Þ�wyZ��_b\=Z]X�Á��ÂÃy��dbªy~�u��g��aµ¡UZP^`w{_;��a¯uxwy��u�uxwyZP�G_b����_���~�a{�H�]_9ay�y~�u`~�_ba{^��9\`ZP^=�{¶[�]~�Z]a5u�G_b\��9�{X¤~�u`ux~�ay�7�7^`�{\`\`_9�5��uxZP|��9\=~¦�9¡y��Z�ú�ÿcG~Qj�ø ~�a5uxZ}\`�]Z}z{ux^r�9���/�y~�\`Z]�4uxZ}�¨z1�Luxwy^��G\`_9X�ú×ux_¤ù¤Û±�9a{�cG~�~¦j¸��c-�.��� j�~�^�~��yZ]a5ux~�®±�9¡y��ZP~�a¬± ) �stay�yZ}Z]�/ªº~��Ý�]_9ay�y~�u`~�_ba ceI�j�wy_b���{^*uxw{Z]a ��Z��g�9a¹��\=~�u`Z �¤c������ j !�¤c������ � �/?j��Y¡¬Z}�g�9�y^=Z2cDù;:<:HúÝ� ø'j1> ?CB ��ã��{uz��c-�.�4� � �/?j�^=ux�9ay�y^Y�G_b\$uxwyZ��]�9�y^x���Z}­¬Z]�4u�_9�°ø _ba¯ù*~�av�HX[_A�yZ]�U�b_�|bZ}\`ayZ}�µ¡5�¸± ) ª���wy~���wA»Ü¡5���]_bay�{~�ux~�_baæcG~�~¦j4ª~�^P~��yZ]a5u`~�®±�9¡{��Z9�HÞ$\`�9ay^`����uxZ}�æux_¯_b�y\P��wy_9��Z]^euxZ}\`_b�YZ��y��X¤zy��Z9ª°uxw{Z]^`ZF�}_bay�y~�ux~�_9a

\`Z}¢?�y~�\=Z¤uxwy��uHu`wyZ]\=Z�¡¬Z¤ay_¸�y~�\`Z}�}u2Z4­�Z}�}u7_����y~�Z4u2_baMwyZg�9\eu2�}_bay�y~�ux~�_9ay^H��ay�ay_æ�}_ba{�G_b�ya{�y~�ay�«_9�r��wy_b��Z}^=u`Z]\`_9����Z4|bZ]��^7�9ay��w{Zg�9\eu7�y~�^=Zg�9^=Z9ª£�yay��Z]^=^2��Zv�]�9aayZ}�{ux\`�9��~�ä]ZP^`�{��w«�]_9a{�G_b�yay�{~�ay�F¡5�¯�9�y�{~�ux~�_ba1�9�¬X¤Z]�9^`�{\`Z]X[Z]a5u`^]�

è�~��9�y\`Z}^æÁ{�Â÷1cDZgj«�9ay��Á{�Â÷1cGw±j,cD~�a¿�Z]�4ux~�_ba�Á{� ì�� � j¨~������y^eux\x�LuxZ'X¤_A�yZ]��^«~�a��wy~���w7¡¬_9u`w2�]_9ay�y~�u`~�_ba{^$wy_9���/��åÜ~�u`w2è�~��b�y\`Z�Á{�Â÷1cDZgj4ª��G_9\�Z4�{�9X[zy��Z9ª���Z�_b¡�u��9~�auxw{~�^�Z]^eux~�X��9a{�

�¤c������ j�! ��c-�.� � �/?j�! ��c-�^� �4�/?j�¤c ���/bj À cÀÁ{�ýÃ?ìbj

Page 36: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d�ÁbfyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�ÊÞ�wy~�^r�]�9aµ¡¬ZPZ}^=u���¡y��~�^`wyZ}�¨�y~�\`Z}�}ux����¡5��®1\=^=u©�9zyzy���A~�ay�Fê��y��Z�Á2u`_¤���y� �/yª

�¤c������ jz! ��c-�.�4� � �/?j�¡¬Z}�g�9�{^`Z�cÀùF:<:HúÝ� ø'j1> ?CB ��9ay�¯uxw{Z]a¨�9zyzy����~�ay�Fê��y��Z � ux_[Z4����w1��ay�bZÓ� ��~�u`w ÿ

��c-�.�4� � �/?j�!��¤c���� �¸�/?j�¡UZ]�]�9�y^`Z�cÀù;:G:Pø¤� úoj#>e? B�À½©�}�]_b\=�y~�ay�ou`_FcÀÁ��ÂÃ?ì9j4ª9_9ay����_bayZ���Z}|9Z]�{_��¬úã^`�{¶[�]Z}^Æ�G_b\$u`wyZ�~��{Z]a5ux~�®1�g��u`~�_ba

_9�1�¤c������ j��G_b\2�9a5��|9�����yZ}^2_9�A���9ay� ��sta¾_9uxwyZ}\H��_b\`�y^}ª�ú�ayZ]Z}�¼a{_9u7¡UZ|��9\`~�Z]�¾��u2�9���ÀÛ£~�u2�]�9a�^`~�X¤z{���'¡¬Z�wyZ]�����]_bay^eu��9a5uF¡5�,Z��AuxZ]\=a1�9��X[Zg�9a{^7�9ay�/ª~��$uxwyZ��9^=^`�yX[z{ux~�_bay^rZ]X2¡U_A�y~�Z}�¨~�a�±"�9\=ZS|��9��~��/ªyu`wyZS\]�ýw/�Â^}�¬_���cÀÁ��ÂÃ?ì9j�^=wy_b�y�����u=u��9~�a¸uxwyZF^x��X¤Z2|��9���yZ2\`Z}�5�9\=�y��Z}^`^�_9�¥uxwyZ¨cD�}_bay^eu��9a5u�jo��Z}|bZ}�£��uo��wy~���w,ú ~�^¡UZ]~�a{��wyZ]���/�osta¸z{\x�9�4ux~��}Z9ª/w{_;��Z}|9Z]\}ª°^`Z}|9Z]\`�9�Y��Z4|bZ]��^©_9��ú���~�����¡UZHayZ}Z]�yZ}��u`_Z]a{^`�y\=ZFu`w1��uPZ}ay_b�y�9w'^`�9X[zy��Z}^P�9\`ZH_b¡{u���~�ayZ}�¸�G_b\oZg����w'�{Z]^`~�\`Z}��|��9���yZ2_9�£ø��è1_b\PZ4�{�9X[zy��Z9ª°~��£��Z���\`Z7~�a5u`Z]\`Z}^=u`Z]�M~�aæuxwyZ7�y~�­�Z}\`Z}ay�]ZTG�cDù¬��� j0H�G�cÀùâ��� = j4ª��wyZ}\`Z �9ay� = �9\=Z�ut��_æu`\`Zg�LuxX[Z]a5u7��Z4|bZ}��^]ªYu`wyZ]a���Zv^`wy_b�{������wy_A_b^`Z�ut��_|��9���{Z]^A/���ay�¬/ = _��Yú×uxwy��u�X��;�{~�X¤~�ä]ZruxwyZ�a?�yX2¡UZ]\�_9�$^x�9X[zy��Z]^�~�a �9ay� =cD\=Z]^=z¬Z}�}ux~�|bZ}���{jr��ay�vuxwyZ}a¨Z}^=ux~�X��LuxZG�cDù¬��� j·HÈG�cÀù���� >= j�!]G�cÀù�� ���/?jÕHÈG�cÀù¬� >= ���/ = j�Àí¯î=f g lYp;hji ö q�p �lk¼mO� ò � óFn ð ô¸m ñ�òYönm ppo ö � ö}ò\qè�~��b�y\`Z Á{�M��^`w{_;��^�^=~�X[zy��Z��y~��9�b\`�9X¤^¥~�a[��wy~���w��¤c������ j¥�]�9ayay_9u�¡¬Z ~��yZ}a?u`~�®1Z}�_���~�ay�Sux_Hu`wyZ©z{\`Z]^=Z]ay�}ZP_9��� àe¡¬_��AáHzy��u`u`Z]\`a?»!�H�}_ba{�G_b�ya{�y~�ay�H�9\`�2cD�1��^`wyZ}�±jZ]XH¡y\x�9�}~�ay�¯�v�]�9�y^x���Y��~�a{´¨¡UZ}ut��Z]Z}aMø �9ay��ù¤�2½Ú�}_ba{�G_b�ya{�y~�ay�¯��\`�F\`Z}zy\`Z��^`Z}a5ux^Hu`wyZ�Z4��~�^=uxZ}ay�]Z�~�a�u`wyZ��{~¦�9�b\`�9X _9���¨¡1�9��´b�t�y_A_b\Sz1��u`wMuxw1�Lu2�]_9a?ux�9~�a{^_bay���[�yay_b¡y^=Z]\e|bZ]�µ|��9\`~��9¡y��Z}^���ay�vw1�9^�ay_H�]_ba5|bZ}\`�b~�ay�[�9\`\=_���^]�£è1_b\�Z��{�9X¤z{��Z9ªuxw{Z¸z1��u`wãøÇ�µúYü� �Ñ�vú�v~�aÝè�~��9�y\`Z¨Á{��d¸�g�9aÝ¡UZ«\=Z]zy\=Z]^=Z]a5uxZ}�×�9^��M�]_baA��G_b�ya{�y~�ay�S�9\=��¡¬Z4ut��Z}Z]a�ø&�9ay��ú�g�ƽܡ¬_��ãz1��u`u`Z]\=a¤\=Z]zy\=Z]^`Z}a5ux^���a¤Z}¢A�y��ux~�_ba��!�»_Ð�c ��Ô��:¾3Ð�j�ªA��w{Z]\`ZèÒ×~�^��yay_b¡{^`Z]\e|bZ}�«�9a{�v�yZ]zUZ]ay�{Z]a5u�_ba�ø��£¿A�y��wµ�9aZ]¢?�1�Lux~�_9a«�y_AZ]^ ay_9u z¬Z}\`X[~�u�uxwyZS~��{Z]a5ux~�®1�g��u`~�_baµ_9�Y�]�9�y^`�9��Z}­¬Z]�}u`^]ª/^=~�ay�}ZH��a?�zU_b\=u`~�_ba¨_9�YuxwyZ�_b¡y^=Z]\=|9Z]�'�yZ]zUZ]a{�yZ]ay�}Z7¡¬Z4ut��Z}Z]a�ø �9ay�¸ù�X��g�¨�9�����g�A^o¡UZ��u=ux\`~�¡y�{u`Z]�vux_F^`z{�y\`~�_b�y^��yZ]zUZ]a{�yZ]ay�}~�Z}^©X[Z]�y~���uxZ}�v¡?�ÇÒ��

Þ�wyZozy\=Z]^=Z]ay�}ZS_����H¡U_;�!z1�Lu`uxZ}\`a�z{\`Z}|9Z]a5ux^�u`wyZ©~��yZ]a5ux~�®1�g�Lux~�_9a�_9���¤c������ jZ}|9Z]a¼��wyZ]aÍ~�uF~�^2�G_9�yay�¼~�a�uxwyZ¯�}_ba5uxZ4�Au¤_9�©�'���9\`�bZ}\2�b\`�9zyw/ª��9^F~�a¼è�~��b�y\`Z

Page 37: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎsr?κÔ�Ë�ÇoȣŠÓ4Ä�Ç©Ñ!É$Ê¥Ð�É�ÐMÙ�Ø�ÓOÒPÊ£ÖFÉ�ÓOØ�ÓtÇ/í�ÓOÑ°Ó]É�t d�Á9÷Á{�M�ycD¡1j4�7Þ�wy~�^P~�^P~�a��}_ba5ux\x��^=u�ux_¯��~�ayZg��\oX¤_A�yZ}��^]ª¬��wyZ]\=Z[u`wyZ[�9�y�y~�ux~�_9a¸_9���9a�9\=�µu`_���¡U_�� z1��u`u`Z]\=aã�g��a+\`Z}ay�yZ]\Ñ��c-�.�4� j7~��yZ}a?u`~�®±��¡y��Z�cD^=Z]Z · wy�9z{uxZ}\vì�ªè�~��b�y\`Z¨ì��Â÷5j��Ýè1_b\FZ4�{�9X[zy��Z�ª�~��où ~�^[\=Z]����uxZ}�Íux_'ø |A~��M����~�a{Zg�9\F\`Z}�¦��u`~�_ba�Ñ! æ �¢Ô�ª±��wyZ]\=Z Ò*~�^ �9a¸�yay_9¡y^`Z}\=|bZ}���y~�^eux�y\=¡1�9ay�}ZHzU_b^`^=~�¡y���¯�]_b\=\`Z]����uxZ}���~�u`w¤ø�ª?uxw{Z]a¸æA! uu

7 G�cÀù���� jY~�^¥ay_9u�~��yZ}a5ux~�®1�9¡y��Z���³r_;��Z}|9Z]\}ª1�9�y�y~�ay�2��a��9\=�ú c ø u`_«uxwyZ¤^=u`\`�y�4ux�y\=Z'cD~J�ýZ9��ª$®yay�y~�a{�æ�¨|��9\`~��9¡y��Z�ú u`w1��uH~�^S�]_b\=\`Z]����uxZ}���~�u`wMø ¡y��u2ay_9u���~�u`w¢Òoj���_b�y���,�D���]~���~�ux��uxZ2u`wyZ��}_bX¤z{�{u���u`~�_ba�_9� G�cDù¬��� j|A~¦�vuxwyZ[~�a{^=ux\=�yX[Z]a5u��9�Æ|��9\=~¦�9¡{��Z2�G_b\=X2�y�¦��c�ã�_����yZ]a,�9ay�'Þ$�y\=´?~�ay�9u`_baãdg÷bf9ÃyÛ^`Z}ZH����^`_ · wy�9z{uxZ}\oì9j4ÿ

æE+! vv G�cÀù���� j�! G�cDù¬� /5jG�cQø¤�</bj !xw ÐJ)

w Ï*) À cÀÁ{�ýÃ5h5j

staºay_ba{z1�9\x��X¤Z4ux\`~���X¤_A�yZ]��^]ª��9�{�y~�ay�ã�9a×~�ay^=u`\`�yX[Z]a5ux�9�P|9��\`~¦��¡y��Z�ú u`_��¡¬_���z1��u`u`Z]\=a cDè�~��b�{\`Z�Á{�M�ycD¡1j`j��y_AZ]^¯ay_9u¯zUZ]\=X¤~�u�uxw{Z'~��yZ]a5u`~�®1�]��ux~�_baÝ_9��¤c������ j��¤Þ�wy~�^�~�^S�¯�D��X¤~���~¦��\ zy\=_b¡y��Z]X�~�a'u`wyZ���a1�9���A^=~�^P_9���]��~�ay~��]�9�$u`\`~��9��^P~�a��wy~���w'u`\`Zg�LuxX[Z]a5uS�9^`^=~��ba{X¤Z}a?u¤cÀúoj©~�^P\`�9ay�y_9X¤~�ä]Z]�¼cGwyZ]a{�]Z9ª�ay_¯��~�ay´µZ]a5uxZ}\`^úoj�¡y�{uo�}_bX[zy��~��9ay�]Z�~�^©~�X[zUZ]\=�GZ}�}u2cD^=Z]Z · w1��z{uxZ}\of5j��©Þ�wyZ2�}_ba{�G_b�{ay�y~�a{���9\=�¡UZ}ut��Z]Z]a7ø��9ay�2ù!~�aHè�~��b�{\`Z¥Á{�M�ycD¡±j°\`Z}zy\`Z}^`Z}a?u`^Y�yayX[Zg�9^=�y\x��¡y��Z¥�D�9�}u`_b\`^�uxw1�Lu~�a��1�yZ]a{�]Z�^`�{¡A³=Z]�4ux^]û5��w{_b~��}Z�_9�1ux\=Zg��u`X¤Z}a?u�cGø'j���^���Z}���{�9^Æ^`�{¡A³=Z]�4ux^]û?\=Z]^`zU_bay^=Zux_vu`\`Zg�LuxX[Z]a5u¤cÀùHj4�7sta�^`�y��w'ux\=~¦�9��^]ª/~�uP~�^Pay_�u�z¬_b^=^`~�¡y��ZHu`_¯_b¡{u���~�a��9a'�ya?¡y~���9^=Z]�[Z]^=u`~�X¤��uxZ�_9�±uxwyZ�u`\`Zg�LuxX[Z]a5u£Z}­¬Z]�4u��¤c������ j$��~�u`wy_b�{uYX¤�9´?~�a{���9�y�y~�ux~�_9a1�9��9^=^`�yX[z{ux~�_bay^�_9a�uxwyZPa1�Lux�y\=Zo_9��uxwyZ©~�a5uxZ}\x�9�4ux~�_9ay^�¡UZ}ut��Z]Z]aµ�]_9X¤zy��~¦�9a{�]Zo��ay�\`Z}^`zU_bay^=ZæcÀ��^�~�^��y_9ayZ9ª��G_b\�Z4�{�9X[zy��Z�ª�~�a�uxw{Z¤zU_9uxZ}a5ux~¦�����t_9�{ux�}_bX¤ZF��zyzy\`_b�9��wux_«~�ay^eux\`�{X¤Z}a?ux�9��|��9\=~¦�9¡{��Z]^��yZ}|9Z]��_bz¬Z}��~�a,stXH¡¬Z}ay^F�9ay��½ ay�b\`~�^=u�d�÷b÷9Ãæ��ay�½©a{�b\`~�^=uFZ}u¤�9�À��d�÷b÷9h{�¾½©��u`wy_b�y�9w¾u`wyZµ�9�y�yZ}�+�9\=�µú c ø z¬Z}\`X[~�ux^7�{^Fu`_�g�����]�{�¦��u`ZP¡¬_b�{ay�y^�_baâ��c-�.��� j�cÀê�_b¡y~�a{^Pd�÷bfb÷{Û{^`Z]���$d]�yÛ±§«�9a{^`´?~�dg÷b÷bi{Û>ã��9��´�Z�9ay�Ü´�Zg�9\=�od�÷b÷��bjH�9ay�,u`wyZv�yzyzUZ]\F��ay����_���Z]\7¡U_b�yay�y^7X¤�g�MZ4|bZ}aÍ�}_b~�ay�}~��yZ�G_b\ �]Z]\eu��9~�a«ut�Az¬Z}^P_9�£�y~�^=ux\=~�¡y��ux~�_9ay^õ�¤c �Z�^�:/bjHcJ¿AZ]�}u`~�_ba¸f{� � �ÂÃ?j�ª¬u`wyZ]\=Z2~�^ ay_������_����]_9X¤zy��ux~�a{����c-�.��� j©�G_b\�04à904ß�37z¬_b^=~�u`~�|bZ7�{~�^=u`\`~�¡y�{ux~�_baÄ�¤c �����:/5j�ª$��^\`Z}¢?�y~�\=Z]�µ¡?�¬²oZ4®1ay~�ux~�_9avÁ{� � �ÂÃ{�

sta¤�9Z]ayZ}\x�9�ÀªbuxwyZr�9�y�{~�ux~�_ba2_9���9\=�]^Yux_S�P�g�9�{^x�9�1�y~��9�b\`�9X*�]�9a[~�X[z¬Z}�yZ9ª5¡y��uayZ4|bZ]\$�9^=^`~�^eugª�uxw{Z¥~��yZ]a5u`~�®1�]��ux~�_bao_9���g�9�{^x�9�bZ4­�Z}�}u`^$~�aPay_bayz1��\x�9X[Z}u`\`~���X¤_A�yZ]��^]�Þ�wy~�^Y~�^Y¡UZ]�g���y^`Z�^`�{��w��9�y�{~�ux~�_ba2\=Z]�y�y�}Z]^£uxwyZ�^=Z}u¥_9�^ê9�t^`Z}z1�9\`��ux~�_ba7�]_ba{�y~�u`~�_bay^�g��\`\`~�Z]�¸¡5�µuxwyZH�y~¦�9�9\x�9XµÛ±wyZ]ay�}Z9ª°~��£���g���y^x�9�$Z}­¬Z]�4uP�yZ]\=~�|���u`~�_baµ�D�9~���^r~�a«uxw{Z_b\=~��b~�a1�9�U�y~¦�9�9\x�9Xµª?~�u�~�^�¡¬_b�{ay�vux_2�D�9~���~�a�u`wyZ��9�y�9X¤Z}a?u`Z]�¯�y~��9�b\x��X��9^���Z]���J�· _ba5|bZ}\`^=Z]���9ª{��a?�H�g�9�{^x�9�yZ4­�Z}�}u¥�yZ}\`~�|���u`~�_baSuxw1�Lu£^`�y�}�]Z]Z}�y^¥~�a2uxw{Z��9�y�bX[Z]a5uxZ}�

Page 38: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

dgÃ5iyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê�y~��9�b\x��X cG¡5�¾�¸^`Z}¢A�{Z]ay�}Z«_9�r^=�AX2¡U_b��~��¤u`\x�9a{^=�G_b\=X���u`~�_ba{^]ª£�9^7~�a · _b\`_b���¦�9\e�Á{�ýÃy� � j���_9�y���¯^`�{�]�]Z}Z]�¸~�avuxwyZ�_b\=~��b~�a1�9�U�y~¦�9�9\x�9Xµ�

X Y

(a)

Z

X Y

(b) (c)

Y

X

Z1 Z2

è�~��b�y\`ZHÁ{�M��ÿ�cÀ�5j�½�¡U_;�z1�Lu`uxZ}\`a/ÿ ���]_9a{�G_b�yay�{~�ay�v��\`�HZ}X2¡y\`�9�]~�ay�v���]�9�y^x�����~�ay´¨ø c ù¤ª�uxw?�y^Szy\=Z}|bZ}a5ux~�a{�¨uxw{Z[~��yZ]a5u`~�®1�]��ux~�_ba�_9���¤c������ jPZ4|bZ}aM~�a�uxw{Zzy\=Z]^`Z}ay�]Z�_9���9a¨~�ay^=u`\`�yX[Z]a5ux�9��|��9\`~��9¡y��Z�ú�ª±�9^�~�a�cG¡±j4� cD�gj�½¡U_�����Z}^`^��b\x��zywuxwy��u�^=u`~�����zy\`_bw{~�¡y~�ux^�u`wyZ�~��yZ}a?u`~�®1�]��ux~�_bav_9�Â��c-�.�4� j4�

²o�y\��9¡{~���~�ut�,ux_M�]_bX[zy�{u`ZÇ�¤c����0��� j[�9ay�Ã�¤c��5�;��� j7�G_b\¤zy�9~�\=^æcÀùE����ùw��j7_9�^`~�ay�b��Z}ux_9aµ|��9\=~¦�9¡y��Z]^��y_AZ]^ray_9u Z]a{^`�y\=ZH_b�y\r�9¡{~���~�ut��u`_¤�}_bX¤z{�{uxZ�³=_b~�a5ur�y~�^=ux\=~��¡y�{u`~�_ba{^]ªA^`�y��wv��^��¤c��������;�5��� j4�Æè�~��b�{\`Z Á{�)�{cD��j�ª5�G_b\�Z��{�9X¤z{��Z9ªA^`w{_;��^����]�9�y^x����y~��9�b\x��Xá��wyZ}\`Z¥¡¬_9u`wè�¤c4/5���� j°�9ay�,��c�/�;��� j°�9\=Z£�]_bX[zy�{ux�9¡y��ZÆ�9Z}uÂ��c�/5���:/�;��� j~�^�ay_�ug� · _bay^`Z}¢?�yZ]a5ux���bª���Z[�g�9aya{_9uS�]_bX[zy�{u`ZÑ�¤c������ j4��sâuS~�^P~�a5uxZ}\`Z]^eux~�ay�µu`_ay_9u`Z¤u`w1��uHu`wy~�^S�y~��9�b\x��X ~�^�uxwyZ¤^`X¤�9����Z]^=u��b\`�9zyw�u`w1��uH�y_�Z}^Hay_�u2�]_9a?ux�9~�aM�¡U_;�×z1��u=uxZ]\=a«��ay�¯^=ux~����/z{\`Z]^=Z]a5ux^ �9aµ�yay�}_bX¤z{�{u��9¡{��ZP�g���y^x�9�°Z4­�Z}�}u]�

½©ay_�uxwyZ}\H~�a5u`Z]\`Z}^=u`~�ay�µ�GZg��u`�y\`Z¤�yZ]X[_bay^eux\x�LuxZ]��¡5�,è�~��b�y\=Z¤Á��)�ycG��jP~�^�uxw1�Lu�]_9X¤zy��ux~�a{�¨uxw{Z�Z}­¬Z]�4u7_9���7³=_b~�a5u�~�a5uxZ}\=|bZ}a5ux~�_9a,~�^S_9�QuxZ}a,Zg��^`~�Z}\�u`w1�9a,�}_bXF�zy�{u`~�ay�µuxwyZ¤Z}­¬Z]�4ux^2_9��~�ux^��]_ba{^=ux~�ux�yZ}a5u2^`~�ay�b��Z}ux_9a�~�a?u`Z]\e|bZ]a5u`~�_bay^}� � ³rZ]\`Z�ªÆ~�u~�^Hz¬_b^=^`~�¡y��Z[ux_æ�]_9X¤zy��uxZ��¤c������ ���/�4jH�9ay�Ü�¤c������ ���/5�=j4ªY�bZ4u7uxwyZ}\`Zv~�^2ay_¸�����_9�©�]_bX[zy�{u`~�ay�Ó�¤c������ j��Ýè1_b\7Z4�{�9X[zy��Z�ª�u`wyZµ�G_9\`X[Z]\F�g�9aÍ¡¬ZµZ}|��9���1��u`Z]�¼¡?�~�a5|9_b´?~�ay�[ê��y��Z � ~�a¬± Ï*) * ªy�b~�|�~�ay�

��c-�.��� ���/��jÎ! #)$ &O��c-�.� /5�:��� ���/��j3�¤c4/5�2�4� ���/�4j! #)$k&O��c-�.� /5�:� ��/��j3�¤c4/5�2� j�À cÀÁ{�ýÃs�bj

³ _���Z}|9Z]\]ª¬ê��y��Z � �g�9a{ay_9ur¡¬ZS�y^=Z]�¨u`_F�]_ba5|bZ}\=u[��c�/5�2��� �:/��j�~�a5ux_Ý��c�/5�G� �:/��j¡UZ]�g���y^`Z�ª���wyZ}aÍ�}_bay�y~�ux~�_9ayZ]��_baÍú���2ø �9ay�Íú��2��\`Z�ê9�t�}_bayayZ}�}uxZ}�Í~�aܱ(Ïy3CEDGL)W^UEK�W�vGa�I�XGhiDiJ^J4I�}z�mK:J4J4FkH0J�L\IiHAv�Sz0K�}�FZW^üVIivGLMH_Wk~UVDGI�D_K�W^U|I�a�P�F3Y�IiX2J|}mK�H0�I�l�J4D_F3W4F�f'I�}�`GXGJ�K:J�L\IiH_WELMHjJ�DGFzfkI�H0J4Fk�0JVI�l�W4F3�0XGF3HiJ�LMK�y�J4a�F3K�J4}�FkH0J�}mK�H_K�h�F3}�F3H0J�Q�üVI�v_L\H_Wb3ñ*�Q�_~0`�{sbk�÷��iT'{

Page 39: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎsr?κÔ�Ë�ÇoȣŠÓ4Ä�Ç©Ñ!É$Ê¥Ð�É�ÐMÙ�Ø�ÓOÒPÊ£ÖFÉ�ÓOØ�ÓtÇ/í�ÓOÑ°Ó]É�t dgÃ{dcGu`wy\`_9�y�bw7uxwyZ��y�9^`wyZ}�¤��~�ayZ}^�j��$½Ý�bZ}ayZ]\`�9�1�9z{zy\`_5����wFu`_��]_bX[zy�{u`~�ay�©u`wyZ�Z}­¬Z]�4u_9�°³=_b~�a5u�~�a?u`Z]\e|bZ]a5u`~�_bay^S~�^P�{Z}|bZ}��_bzUZ]��~�aÓ´$Z]�9\`�£��ay�Mê�_b¡y~�ay^¤ced�÷9÷5ìbj4Û°u`wy~�^P~�^�yZ}^`�]\=~�¡UZ]�µ~�a · w1�9z�uxZ]\rècJ¿�Z}�}u`~�_ba¯Ãy�ýÃ?j4�

�m�|{��3� «o¦O§.­�®�°'ª=�j°k­?y £ ¥1¦O§.¨'¡è�~��b�y\`Z©Á{�ÂfH^`wy_���^�^=~�X[zy��Zr�y~¦���b\x�9X[^¥~�a���wy~���w�uxwyZP�]�9�y^`�9��Z4­�Z}�}u�_9��ø _9a¯ù~�^©~��yZ}a?u`~�®±��¡y��Z�cG��wyZ}\`Z7ø �9ay�'ù��9\=Z7^`~�ay�b��ZS|��9\`~��9¡y��Z}^�j��H¿A�y��w�X[_A�yZ]��^o�9\=Z�g������Z}��à=~��yZ}a5ux~��Q�A~�ay��áv¡UZ]�g���y^`Z[uxwyZ}~�\P^=u`\`�y�4ux�y\=Z]^��]_bX[X2�yay~��g��u`ZF�µ^=�{¶[�]~�Z]a5ua?�yX2¡UZ]\S_9����^`^`�{X¤z{u`~�_ba{^�cGX¤~�^`^=~�ay�¯��~�ay´?^�j©ux_µzUZ]\`X[~�uPu`wyZ[~��yZ}a?u`~�®1�]��ux~�_ba�_9�uxw{Z�u��9\=�bZ}uY¢?�1��a?u`~�ut���¤c������ j4������u`Z]a5uY|��9\`~��9¡y��Z]^Æ��\`Z�ay_9uÆ^`wy_���a[Z4��zy��~��]~�ux���S~�auxw{Z]^`ZF�y~��9�b\`�9X¤^}Û/\x�LuxwyZ}\]ª°^`�{��w�|��9\`~��9¡y��Z]^o��\`Z7~�X[zy��~��]~�ur~�a¸uxw{Z7�]_ba{�G_9�yay�y~�ay��9\=�]^HcD�1�9^=wyZ]�1j4��WÆ|bZ}\=�µ�g���y^x�9���y~��9�b\`�9X���~�u`w¨����uxZ}a5ur|��9\=~¦�9¡{��Z]^��]�9a«¡UZ��]_baA�|bZ}\=u`Z]��ux_��9a¤Z]¢?�y~�|��9��Z]a5u��y~¦�9�9\x�9X ~�a5|b_b��|�~�ay�SX¤Z]�9^`�y\=Z]��|��9\=~¦�9¡{��Z]^Y~�a5uxZ]\=�]_baA�ayZ}�}uxZ}�,¡5���9\`\=_;��^H�9ay���]_ba{�G_9�yay�y~�ay�¸�9\=�]^}�vÞ�wy~�^S�]_ba5|9Z]\`^=~�_ba��}_b\`\=Z]^=z¬_ba{�y^ux_H^`�y¡y^eux~�ux�{u`~�ay�7_b��u��9���¬����uxZ}a?u�|��9\`~��9¡y��Z}^¥�G\=_bXÚuxwyZo^eux\=�y�}u`�y\x�9�¬Z]¢?�1��u`~�_ba{^�_9�cÀÁ�� � jY�9a{�[u`wyZ]a¤�]_bay^eux\=�y�}u`~�ay���Sa{Z}�Ý�y~��9�b\`�9X�¡5�F�]_9ayayZ]�4ux~�ay�H��a?�7ut��_�|��9\=~���9¡y��Z]^©ø º ��ay�¸øS¡5�¾cG~Qjr�9aæ�9\=\`_����G\`_bX ø�u`_�ø º ��wyZ]ayZ4|bZ}\�ø��9zyzUZg�9\=^~�avu`wyZ�Z]¢?�1��u`~�_ba¯�G_b\�ø º �9ay�McD~�~¦j��7�]_ba��G_b�yay�y~�ay�¤�9\`�o��wyZ}ayZ}|9Z]\©u`wyZ�^x�9X[Z?¾uxZ}\`X ��zyz¬Z]�9\`^�~�a'¡U_9u`w » º �9ay�È»*;�[Þ�w{Z[\=Z]^=�y��uS~�^��v�y~��9�b\`�9X�~�a���w{~���w��9����yayX[Zg��^`�y\=Z]�¯|9��\`~¦��¡y��Z}^��9\`ZPZ��{_9�bZ]ay_9�y^©��ay�¨XH�{ux�1�������¤~�a{�yZ]zUZ]ay�{Z]a5ug�

¿�Z}|9Z]\`�9���GZg�Lux�y\=Z]^7^`w{_b�y���M¡¬Z�a{_9uxZ}�,�G\`_bX Z4�{�9X[~�a{~�ay�µuxwyZ��{~¦�9�b\`�9X[^�~�aè�~��b�y\`Z�Á�� f{�db� ¿�~�ay�]Zou`wyZP\`Z}X¤_�|��9�¬_9�$�9a5�v�9\=�P_b\��9\=\`_��á�G\=_bX��2�]�9�y^`�9�/�y~��9�b\x��X��]�9a_ba{�����9^`^=~�^=u�u`wyZP~��{Z]a5ux~�®±�9¡y~���~�ut�F_����]�9�y^x���/Z}­¬Z]�4ux^]ª��¤c������ j���~����U^=u`~�����¡UZ~��yZ]a5ux~�®1Z]�µ~�aµ�9a5�¯Z}�y�bZS^`�y¡y�9\x�9zyw¨_9��uxwyZS�y~¦���b\x�9X[^�^=wy_���a¸~�aµè�~��b�y\`ZÁ{�Âf{�Ä�°~�´�Z}��~�^`Z9ªÆu`wyZ�~�a?u`\`_A�y�y�4ux~�_9a,_9��X[Z]�y~���ux~�ay�¨_9¡y^`Z}\=|bZ}�¼|��9\=~¦�9¡{��Z]^_ba5u`_2�9a5�FZ]�{�bZ©~�a��S�g�9�y^`�9�±�b\x��zyw��]�9a��9^=^`~�^=ugªA¡y�{u�ayZ4|bZ]\�~�X[z¬Z}�yZ9ª5uxw{Z~��yZ]a5ux~�®±�9¡y~���~�ut��_9�Æ�9a5�¯�g�9�{^x�9��Z}­¬Z]�4ug��Þ�w{Z]\`Z4�G_b\`Z�ª|�¤c������ j���~����°^eux~����/¡UZ~��yZ]a5ux~�®1Z]�[�G\`_bX �9a5�¤�9\x�9zyw¤_b¡{ux�9~�ayZ}�¤¡5�¤�9�y�{~�ay�SX¤Z}�y~¦�Lux~�a{�Sa{_��yZ}^¥u`_u`wyZ��y~¦�9�9\x�9X[^�^`w{_;��a¨~�a¯è�~��b�y\=Z�Á{� f��

� ��Þ�wyZµ�y~��9�b\x��X¤^7~�aãè�~��b�y\=Z¯Á{� f'�9\=ZµX¤�L��~�X¤�9��~�a�uxwyZµ^`Z}ay^`Zµuxw1��u[uxw{Z~�a?u`\`_A�y�y�4ux~�_9av_9�Y�9a5�¯���y�y~�u`~�_bay�9�/�9\=��_b\��9\`\=_��º_ba5ux_¤�9aµZ4��~�^eux~�ay�Fz1�9~�\_9�$ay_A�yZ]^���_b�y���¨\=Z]ay�yZ}\1��c-�.��� j�ay_[��_bay�9Z]\�~��yZ]a5ux~�®±�9¡y��Z9�

Page 40: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

dgà � ÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê

X

Y

(a)

Y

ZX

(d)

X

Y

X

Y

Z

Z1

Z2

Y

(g)

Z3

XX

Y

Z

(e)

X

Y

Z

(b) (c)

Z1

Z2

(f)

è�~��b�y\`Z�Á{� f�ÿ$Þ£�Azy~��]�9�yX[_��yZ}��^Æ~�aH��wy~���w[u`wyZ�Z}­¬Z]�4u£_9�±ø&_9aFùº~�^Æ~��yZ]a5ux~�®±�9¡y��Z9�²P�9^=wyZ]�¾�9\=�]^2\=Z]zy\=Z]^=Z]a5uF�]_ba��G_b�yay�y~�ay�¸z1�Luxwy^}ª£�9ay�¼ú�\=Z]zy\=Z]^`Z}a5ux^F_b¡y^=Z]\=|9Z]��]_�|��9\=~¦��u`Z]^]�Á{��½ ��uxw{_b�y�bw,X[_b^eu7_9��uxwyZv�{~¦�9�b\`�9X[^�~�a¾è�~��b�{\`Z�Á�� f¸�}_ba5u��9~�a�¡¬_���z1��u=�u`Z]\`a{^]ªUay_bayZS_9��u`wyZ]^=ZHz1�Lu`uxZ}\`ay^rZ]X¤�9a1�LuxZ]^��G\`_9X�ø cD�9^ ~�^�uxwyZS�g�9^=ZH~�aè�~��b�y\=Z]^�Á{�Â÷1cÀ�bj��9ay�¤cD¡1j¬ux_��G_b����_�� j4��sta��bZ]ayZ}\x�9�ÀªL��a{Z]�]Z}^`^`�9\=���}_bay�y~�ux~�_9a�G_b\�uxw{Z�~��yZ}a5ux~�®1�9¡y~���~�ut�æ_��A��c-�.�4� jS~�^�u`wyZ��9¡{^`Z]a{�]Zv_9�r�µ�]_ba{�G_9�yay�y~�ay��9\=�P¡¬Z4ut��Z}Z]a«ø �9a{�«�9a5�v��w{~����¯_9��ø uxw1��u�~�^��9aµ�9ay�}Z]^=u`_b\r_9�Æù[�

Ãy�1²o~��9�b\x��X¤^£cD�5j°�9ay��cD¡±j°~�aSè�~��b�y\=Z�Á�� f��]_ba5ux�9~�aSay_r¡1�9��´b�t�y_A_b\�z1��u`wy^�¡¬Z��ut��Z]Z}a�ø&�9ay�¤ù��ay�[u`w?�y^�\=Z]zy\=Z]^=Z]a5u�Z��{zUZ]\=~�X[Z]a5u����1�yZ]^=~��bay^£~�aF��w{~���wu`wyZ]\=ZS~�^�ay_¤�}_ba{�G_b�{ay�y~�a{�[¡{~¦�9^�¡UZ}ut��Z]Z}a«uxwyZPu`\`Zg�LuxX[Z]a5uHcGø'j��9ay�¯uxw{Z\=Z]^`zU_bay^=Z�cÀù7j�Û�wyZ]a{�]Z9ª���c-�.�4� jÝ! ��c-�.� j4�È��~�´9Z4��~�^`Z�ª��{~¦�9�b\`�9X[^¯cD�gj�9a{�McG�±j�~�a¯è�~��b�{\`ZPÁ{�Âf2\`Z}zy\`Z}^`Z}a?u �yZ}^`~��9ay^�~�a���wy~���w¯_b¡y^`Z}\=|9Z]�«�}_�|9��\`~����u`Z]^¯ú�¡y��_A��´¾Z}|bZ}\=�ã¡1�9��´b�t�y_A_b\�zy��uxw+¡UZ}ut��Z]Z}a!ø �9ay�Ýù cG~J�ýZ9��ª�ø~�^ à=�}_bay�y~�ux~�_9a1�9�����æ~��9ay_b\x��¡y��Ziá«�b~�|bZ]a¼úSª¥~�aMuxw{Z��¦�9a{�b�1�9�bZ¤_9� êr_9^`Z]aA�¡1���yX¹��ay�¾êr�{¡y~�aÝd�÷9fbÁ5j4Û£wyZ}ay�]Z�ª��¤c������ jS~�^H_b¡{u���~�ayZ}��¡?�M^=ux�9ay�1�9\=��9�;³=�y^=u`X¤Z}a?u��G_b\rúÚcD�9^�~�a�cÀÁ{��d�÷5j=j4ÿ

�¤c������ j�!¶# $ �¤c���� �:/5jk��c�/5j�À

Page 41: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎsr?κÔ�Ë�ÇoȣŠÓ4Ä�Ç©Ñ!É$Ê¥Ð�É�ÐMÙ�Ø�ÓOÒPÊ£ÖFÉ�ÓOØ�ÓtÇ/í�ÓOÑ°Ó]É�t dgÃbÁì��rè1_b\°Z]�9��w�_��{uxw{Z¥�{~¦�9�b\`�9X[^/~�aSè�~��b�y\`ZÆÁ{� f�ªL��Z£\`Zg���y~����o_b¡{ux�9~�a����G_b\=X2�y�¦��G_b\ �¤c������ jF¡?�Í�y^=~�ay��^=�AX2¡U_b��~��¨�yZ}\`~�|9�Lux~�_9ay^¤zy��u`u`Z]\`a{Z]�Ü���Qu`Z]\Fuxwy_b^=Z~�aÜ¿�Z]�4ux~�_ba!Á{�ÂÃ{� Á{��Þ�wyZæ�{Z]\`~�|���ux~�_ba+~�^�_9�QuxZ}aÝ�b�y~��{Z]�Ý¡5�ãu`wyZæ�b\x��zywu`_bz¬_9��_b�9�9�!è±_9\¤Z��y��X¤zy��Z9ª��y~��9�b\`�9X cG�xj[~�a+è�~��b�y\=Z«Á{�Âf��{~��}ux��uxZ}^�uxw{Z�G_b����_���~�a{�2�yZ]\=~�|���u`~�_ba/�$åM\=~�ux~�ay�

�¤c������ j�!"#$ &V» $ *s��c-�.� /5�:�:/���� j3�¤c4/5���:/���4� j����Z¨^=Z]Zvuxwy��uFuxwyZ¯^`�{¡y�b\x��zywÍ�}_ba5u��9~�ay~�a{� Þ�øÇ��ú�����ú��0߸~�^7~��yZ}a5ux~��]�9��~�a^eux\`�{�}ux�{\`Z7ux_vuxwy��u�_9�¥�y~¦���b\x�9X¹cDZgj4ª/��~�uxwÍcJú�����ú���j©\`Z}zy�¦���]~�a{�'cJúj��ù�j4ª\=Z]^`zUZ]�4ux~�|bZ]���b�ÆÞ�w?�y^}ª��¤c4/5�:�:/����� j��]�9aF¡¬Z�_b¡�u��9~�ayZ]�7�G\`_9X�cDÁ{�ÂÃbÁ5j4�Â��~�´9Z����~�^`Z9ª$uxwyZ[uxZ]\=X �¤c����</5�Z�:/�2��� j��g�9aM¡¬Z¤\`Z]�{�y�]Z}�,ux_ �¤c����</5���:/�� j�¡?�ê��y��Z � ªy^`~�ay�]Z¤cÀùp:G:�ø¤� ú��:��ú���j#> ? �Yå�Z�u`wyZ]\=Z}�G_b\=Z�w1�g|bZ

�¤c������ jÎ! #$ &V» $ *s�¤c����</5�:��/�� j3�¤c4/5�� j6 #S7 9 �¤c4/��� /5��� >= jk��c >= j�À cÀÁ{�ýÃ5f5j

½ zyzy���A~�ay�[�F^=~�X[~����9\��{Z]\`~�|���ux~�_ba�ux_F�y~��9�b\x��X cD�5j�_9�Æè$~��b�y\=ZPÁ{�ÂfF�A~�Z]���{^�¤c������ jÎ! # $ & # $ * #87 9 �¤c����</5���1/�2� >= j3�¤c >= � /�4j

6 ��c�/5�2�</�2� jk��c�/��j cÀÁ{�ýÃ5÷5jér_9uxZ�uxwy��u£uxwyZ�|��9\`~��9¡y��Z�ú���y_�Z}^£ay_9u��9zyzUZg�9\£~�a¸cDÁ{�ÂÃbf5j4ªb��wy~���w�X[Zg�9a{^u`w1��u¯ú��ayZ}Z]�Ýay_�u�¡UZ¸X[Zg�9^=�y\`Z}�+~���������_bayZµ���9a5u`^�ux_,��Zg�9\=aã~�^¤uxw{Z�]�9�y^x���°Z}­¬Z]�}ur_9�$ø _baµù��

h{��sta��y~¦�9�9\x�9X[^[cDZgj4ª�cG�xj4ª$�9ay�ãcD�?jo_9��è�~��b�{\`Z[Á{�Âf{ª�u`wyZ[~��yZ}a?u`~�®±��¡y~���~�ut�µ_9��¤c������ jF~�^[\=Z]ay�{Z]\`Z}�ã�GZg��^`~�¡{��Zvuxwy\=_b�y�bw¼_b¡y^=Z]\e|bZ]�ã�]_�|��9\`~���uxZ}^�ú�uxw1�Lu�9\=Zo��­¬Z]�4uxZ}�µ¡5�[uxwyZ ux\`Z]��uxX[Z]a5u�ø cD^`~�ay�]Z X¤Z}X2¡UZ]\`^�_9�$ú×�9\=Zo�yZ]^=�]Z}a���1��a?u`^�_���ø'j���Þ�wy~�^S^eu��9a{�y^��}_ba5ux\`�9\=�æu`_¨uxw{ZF���9\=ay~�ay�;»Ü\`Z}z¬Z]��uxZ}�M~�aX[_b^=uS_9��uxwyZ[��~�uxZ}\x��u`�y\`Z7_ba�^=ux��ux~�^=ux~��g�9��Z4��z¬Z}\`~�X¤Z}a?ux��ux~�_baA»+ux_¯\`Z4�G\x�9~�a�G\=_bX��9�;³=�y^=u`~�ay�F�G_b\��]_9ay�]_bX[~�ux�9a5u�_b¡y^=Z]\e|9�Lux~�_9ay^�uxw1��u �9\=Z���­¬Z]�}u`Z]�¨¡?�u`wyZ7ux\`Z]��uxX[Z]a5u�c · _g�¼d�÷bì9f{Û$êr_b^=Z]a?¡1�9�{X d�÷bf9ÃyÛe´Y\x��u=u��9ay�M¿���wy�¦��~��GZ]\d�÷9fbf{Û�åM�9~�ayZ]\Pd�÷9fb÷5j4��sâu ~�^��]_9X¤X[_bay����¡UZ]��~�Z}|9Z]�vuxwy��ur�7�]_bay�}_bX[~�u���a?uú u`w1��u�~�^H��­�Z}�}u`Z]�,¡5�'uxwyZ[ux\`Z]��uxX[Z]a5uHX2�y^=uH¡¬Z¤Z4���]���y�yZ}�,�G\`_bX uxw{Z

Page 42: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

dgÃbÃ1ÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê�9ay�9���A^`~�^©_9�£u`wyZ2ux_9ux�9��Z}­¬Z]�}u�_��¥u`wyZ2ux\=Zg��u`X¤Z}a5u[c4´Y\x�Lu`u��9a{�'¿���wy�¦��~��GZ]\d�÷9fbf5j4��Þ�wyZH\=Zg�9^=_ba«�b~�|bZ}a¨�G_b\�uxwyZSZ4���}���y^=~�_ba¨~�^�uxw1��uruxw{Z��]�9���}�y�¦��u`~�_ba_9�{ux_9ux�9�?Z}­¬Z]�}u`^Æ�9X[_b�ya5u`^�u`_©~�a5u`Z]�b\`��ux~�ay� _9�{uYúSª���wy~���w2~�^��G�yay�4ux~�_9a1�9�����Z}¢A�{~�|��9��Z}a5u�u`_«_bX[~�u=ux~�a{�µúÚux_¨¡¬Z}�b~�a���~�u`w/��²o~¦���b\x�9X[^[cDZgj4ª�cG�xj4ªÆ��ay�cG�?j�^`wy_��þ�]�9^`Z}^r��wyZ}\`ZSuxwyZPux_�u��9�°Z4­�Z}�}ux^r_9�$ø �9\=Z�~�ay�yZ}Z]�µuxwyZPu���\`�bZ4u_9��~�a5|bZ}^=u`~��5��u`~�_ba��9ay��ª$Z4|bZ]aM^`_yª�uxwyZ[X¤Z]�9^`�{\`Z]X[Z]a5u�_9���}_bay�]_9X¤~�u��9a5ux^u`w1��u��9\=Zo��­¬Z]�4uxZ}�µ¡5�¤ø cDZ9�ý�y��ª1ú×_b\�ú��`j£~�^�^eux~����±ayZ]�}Z]^=^x�9\e�b��³r_���Z4|bZ]\}ªu`wyZP�9�L³=�{^=uxX[Z]a5u�a{Z]Z]�{Z]�¯�G_b\�^=�y��w¨�}_bay�}_bX¤~�u��9a5u`^�~�^�ay_bay^eu��9a{�1�9\`��ªy~�a��|9_b��|A~�a{�¤ut��_v_b\©X[_b\`ZS^=ux�9�bZ}^o_9�Yuxw{ZH^=ux�9ay�1��\`���9�L³=�{^=uxX[Z]a5u©_9� cÀÁ{��d�÷5jcG^`Z]Z�cÀÁ�� � fbj4ª°cDÁ{�ýÃ5f5j4ªy�9ay�McÀÁ{�ýÃ5÷5j=j4�

����stav�y~��9�b\`�9X¤^ocD¡±j�ª°cG��j4ªy�9ay�'cG�xj�_9�$è�~��b�y\=ZoÁ{�Âf{ªyù*w1�9^��2z1�9\=Z]a5u���wy_b^=ZZ4­�Z}�}u7_ba,ù�~�^�ay_9u2~��yZ]a5u`~�®±�9¡{��Z9ÛYZ4|bZ}a¼^=_yª�u`wyZ�Z}­¬Z]�4u7_9��ø _9a�ù&~�^~��yZ]a5ux~�®±�9¡y��Z9�PÞ�wy~�^o�yZ}X¤_ba{^=ux\`��uxZ}^ouxw1�LuP��_A�g�9��~��{Z]a5ux~�®±�9¡y~���~�ut��~�^oay_�u��ayZ}�]Z}^`^x��\=���]_bay�{~�ux~�_baæ�G_b\��b��_9¡1�9�Æ~��yZ]a5ux~�®±�9¡y~���~�ut�b�Fsta�_�uxwyZ}\S��_b\`�{^]ª$u`_~��yZ]a5ux~��Q��u`wyZvZ}­¬Z]�4uF_9�rø _ba¾ù���Z¯ayZ}Z]�¼ay_9u7~�ay^`~�^=u7_9a�~��yZ}a?u`~��Q�A~�a{�Z]�9��w«��ay�¨Z4|bZ}\=�¯��~�ay´�����_bay�2u`wyZ�z1��u`wy^��G\`_bX ø u`_¤ù[�

�m�|{���� ' ¥�­O°'¦O§.­�®>°'ª}�õ°k­?y £ ¥A¦O§e¨'¡è�~��b�y\`Z�Á{�Â÷�zy\=Z]^`Z}a5ux^Put�Azy~��]�9�$�{~¦�9�b\`�9X[^ ~�a¨��wy~���w¸uxwyZHu`_9u����$Z4­�Z}�}uP_9�Yø _baù¤ªe�¤c������ j�ª�~�^�ay_9uS~��{Z]a5ux~�®±�9¡y��Z9�Fé _�uxZ}��_b\euxw5�æ�GZg��u`�y\`Z}^�_���u`wyZ]^=Z��y~��9�b\x��X¤^�9\=Z���^��G_b����_;��^}�db��½ ���±�b\x��zywy^�~�a�è�~��b�y\=Z©Á{�Â÷H�}_ba5u��9~�a��ya?¡y��_���´L�9¡y��Z©¡1����´5�â�y_A_b\�z1�Luxwy^�¡¬Z��ut��Z]Z}aµø �9a{�¨ù[ª{uxwy��u�~�^}ª{z1�Luxwy^�Z}ay�y~�a{�2��~�uxw¯��\`\`_���^�zU_b~�a5u`~�ay�Hu`_2øu`w1��uS�g�9a{ay_9u�¡¬Z[¡y��_A��´9Z}�M¡5�'_b¡y^=Z]\=|9Z]��ay_9ay�yZ]^=�]Z}ay�1�9a5ux^H_9��ø���Þ�wyZzy\=Z]^=Z]ay�}Z¨_9� ^`�{��w+�æz1��u`w¼~�a¼���9\x�9zyw¾~�^]ª£~�ay�{Z]Z]��ª��æayZ]�}Z]^=^x�9\e��uxZ]^eu�G_b\Ya{_bay~��{Z]a5ux~�®±�9¡y~���~�ut��cD^=Z]Z�Þ�wyZ}_b\`Z}X�Á�� Á{� � j���Þ�w1��u¥~�^¥a{_9u¥�S^=�{¶[�]~�Z]a5uu`Z]^=u¯~�^��yZ}X¤_ba{^=ux\`��uxZ}�Ý¡5�Üè�~��9�y\`Z«Á{� fycDZ�j�ª ~�aã��wy~���wÜu`wyZæ¡1�9��´b�â�y_�_9\z1�Luxw�cD�1��^`wyZ}�±j�~�^��ya?¡y��_A��´L�9¡y��Z��9ay�¯�bZ}uõ�¤c������ j�~�^�~��yZ}a?u`~�®±��¡y��Z��

� ��½�^=�{¶[�]~�Z]a5uF�]_ba{�y~�u`~�_baM�G_9\�uxw{Zvay_bay~��yZ]a5ux~�®±�9¡y~���~�ut�æ_9�j��c-�.�4� j�~�^Huxw{ZZ��{~�^=u`Z]ay�}Z�_9�±�©�}_ba{�G_b�{ay�y~�a{�©z1��u`w2¡UZ}ut��Z]Z]a2ø��9ay�7�9a5�S_9�1~�ux^���wy~����y\`Z}a_ba��vz1��u`w'�G\=_bX�ø ux_¨ù¤ª��9^�^`wy_���a,~�a'è�~��b�{\`Z]^�Á�� ÷1cG¡±jP�9ay�ãcD�gj4�[½^eux\`_9ay�bZ]\r^`��¶¤�}~�Z}a?u �]_9ay�y~�u`~�_ba¯~�^ru`w1��u�uxwyZS�b\`�9zywµ�]_ba5u���~�a¨�9a5�¯_9�Æuxw{Zz1�Lu`uxZ}\`ay^�^=wy_���a«~�aµè�~��b�y\=ZPÁ{�Â÷¤�9^���a¨Z}�y�bZ�^`�y¡{�b\x�9z{w/�

Page 43: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ�~5ÎãÒoÓeÐ�ÄPÏ�ÐyÐ�Ó4Ù�Ö dgÃ5ì

(e)

ZX

Y

Z1Z2

X

Y

(g)(f)

Y

Z

X

Y

(h)

Z

W

X

X

Y

(a) (b)

X

Y

Z

X

Y

Z

(c) (d)

Y

ZX

è�~��b�y\=ZPÁ{�Â÷{ÿYÞ¥�Azy~��g�9��X[_��{Z]��^�~�av��wy~���wâ�¤c������ j�~�^�ay_9u�~��{Z]a5ux~�®±�9¡y��Z9�Á{�j�P\x�9z{wvcD�5j°~�a2è�~��b�y\=Z�Á�� ÷HcD^x��X¤Z��9^$è�~��b�{\`Z¥Á{�M�ycD�gj`j°�yZ]X[_bay^eux\x�LuxZ]^�uxw1�Lu��_��]�9�$~��yZ]a5u`~�®±�9¡{~���~�ut�¯~�^oa{_9uP^`��¶¤�}~�Z}a?u��G_b\ �b��_9¡1�9��~��yZ]a5u`~�®±�9¡{~���~�ut�b�Pè1_b\Z��y��X¤zy��Z9ª{��Z��g�9a¯~��yZ]a5ux~��Q�¸�¤c4/5���� j��A�¤c4/�;�4� j��1�¤c������/5�ej�ª1�9ay�¬�¤c������/�xj¡y��u�ay_9uA�¤c������ j4�YÞ�wy~�^�~�^�_bayZo_9��u`wyZoX¤�9~�a¤�y~�­¬Z]\=Z]ay�}Z]^�¡UZ}ut��Z]Z]aµay_baA�z1��\x�9X[Z}u`\`~��o�9ay�µ��~�a{Zg�9\�X[_A�yZ]��^]Ûy~�a¯u`wyZ��¦��u=uxZ}\]ª1�9�����]�9�y^x����Z4­�Z}�}u`^ �]�9a¡UZ¸�yZ4uxZ]\=X¤~�ayZ]�Ý�G\=_bX u`wyZæ^=u`\`�y�4ux�y\`�9�o�}_�Z4¶¤�}~�Z}a?u`^¯�9ay�ÜZg����w!�}_�Z4¶7��}~�Z]a5u�\`Z]z{\`Z]^=Z]a5ux^�u`wyZ©�]�9�y^`�9�¬Z4­�Z}�}u�_��°_bayZr|9��\`~¦��¡y��Z�_ba¤~�ux^�~�X[X¤Z}�y~¦�LuxZ^=�y�]�}Z]^`^=_b\]�

í¯î}� � ö �Æqrõ ��� ö ó ñ�m���m�3� Ñsx[��¨3°>Ïg~V�®>°'¥�­[¡¤��­[¦Ýz��A®�§e­[¡�°k¥�­[¡Þ�wyZSX¤Z4uxwy_A�y^��yZ4|bZ]��_bzUZ]�«~�avuxwy~�^r��wy�9z{uxZ}\r�D���]~���~�ux��uxZruxwyZS�y\x�g��~�ay�F_9�Æ¢?�1�9aA�ux~�u���u`~�|9Z2�g�9�{^x�9�Y~�a{�GZ]\=Z]ay�}Z]^P�G\`_bX��v�]_bX2¡{~�a1��u`~�_ba«_9��¢?�1����~�ux��ux~�|bZ��g�9�y^`�9�£�9^O�^`�{X¤z{u`~�_ba{^�cGZ]ay�}_��{Z]�2~�aSu`wyZ��{~¦�9�b\`�9X�j/�9ay�Ha{_bayZ4��zUZ]\=~�X[Z]a5u��9�?_b¡{^`Z]\e|���ux~�_bay^]�Þ�wyZ¸�]�9�y^`�9�o�9^=^`�yX[z{u`~�_bay^¤~�a¼uxwyZ}X¤^=Z]��|9Z]^v�]�9ayay_9u��bZ]a{Z]\x�������¾¡UZ¨u`Z]^euxZ]�Ü~�aay_ba{Z4��z¬Z}\`~�X¤Z}a?ux�9��^=u`�y�y~�Z}^]ª��yay��Z]^=^µu`wyZ}�!~�X[zU_b^`Z��]_9ay^=u`\x�9~�a?u`^¨_9aáuxw{ZM_b¡��

Page 44: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

dgÃ5hyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê^`Z}\=|9Z]�«�{~�^=u`\`~�¡y�{ux~�_bay^}�ÆÞ�wyZPX[_b^=u��}_bX¤X[_ba�ut�AzUZ�_9���]_bay^eux\`�9~�a5ux^r�9z{z¬Z]�9\`^�~�auxw{Z��G_b\=X�_��Y�]_ba{�y~�u`~�_ba1���°~�ay�yZ}z¬Z}ay�yZ}ay�]~�Z]^]ª/�9^©�}_bX¤XH�yay~��]��uxZ}�¨u`wy\`_b�{�bw«uxw{Zê9�t^=Z]z1�9\`��ux~�_ba'�]_bay�{~�ux~�_bay^�~�a�uxw{Z¤�y~��9�b\`�9X¤^}��½ ay_9uxw{Z]\�ut��zUZ�_����]_ba{^=ux\`�9~�a5u`^u���´9Z]^7u`wyZ��G_b\=X¹_9��a?�yX[Z]\=~��g����~�a{Z]¢?�1�9��~�ux~�Z]^}�æsta · wy�9z{uxZ}\Ff{ªY�G_b\HZ��{�9X¤z{��Z9ª��Z�^`w{_;�"u`w1��uSuxwyZ¤�9^=^`�yX[z{ux~�_bay^S�9^=^`_A�]~¦�LuxZ]�'��~�u`wM~�a{^=ux\=�yX[Z]a5u��9�Æ|��9\=~¦�9¡{��Z]^cÀè$~��b�y\=ZPÁ{�M�ycD¡1j`j��9\`ZP^=�y¡A³=Z}�}urux_7�D�9��^=~�®1�]��ux~�_ba�uxZ}^=u`^r~�avuxwyZP�G_b\=X�_9�$~�ayZ}¢?�1�9���~�u`~�Z}^ _baæ�}_bay�y~�ux~�_ba1�9�°zy\=_b¡1�9¡{~���~�ux~�Z]^�c4´�Zg�9\=��d�÷b÷bì�¡±j4�o¿Aux~����ÀªU^=�y��w��]_ba{^=ux\`�9~�a5u`^zUZ]\`X[~�u¥uxw{Z©uxZ}^=ux~�ay�2_9�/X¤Z}\`Z]���¤�H^`X¤�9���1�G\`�9�}u`~�_ba�_��/uxwyZ©�g�9�y^`�9�/��^`^`�{X¤z{u`~�_ba{^Z]XH¡¬_A�y~�Z}��~�a¤uxw{Z©�y~��9�b\x��X¤^}Û?u`wyZ©¡{�y��´F_9�/u`wy_b^=Zo�9^=^`�yX[z{ux~�_bay^�X2�y^=u�¡¬Z ^`�y¡A�^=ux�9a5ux~���uxZ}�¨�G\`_9X��{_bX���~�a¯´?ay_�����Z]�{�bZ7�9^ _b¡{u���~�ayZ}�¨�G\`_9X�Z]~�u`wyZ]\�uxw{Z]_b\=Z}ux~��g�9��]_9ay^`~��yZ]\`��ux~�_bay^�cDZ9�ý�y��ª$u`w1��uS�D�9����~�ay�v¡1��\`_bX[Z}u`Z]\`^S�y_¸a{_9uH�]�9�y^=Z�\x��~�a±jo_b\S\`Z���¦�LuxZ]�ÍZ4��zUZ]\`~�X¤Z}a5u��9�r^=u`�y�y~�Z}^]�þè1_b\[Z4�{�9X[zy��Z9ª�uxw{Z«Z4��zUZ]\=~�X[Z]a5u��9�r^=u`�y�{�Í_9�§¨_AZ]\euxZ]�YZ4u��9�J�Æced�÷bfbìbj4ª/��wy~���w�\=Z}�G�{u`Z]��uxwyZFw5�Az¬_�uxwyZ}^`~�^�uxwy��uP|A~�u���X¤~�a · ~�^Z}­¬Z]�4ux~�|bZ¨�9�5�9~�ay^=u[�g��ay�]Z}\]ª��]�9aÍ¡UZµ�y^`Z}�+�9^¤�'^=�y¡y^eu��9a5ux~�|bZ«�9^`^=�yX¤z�ux~�_9aÍ~�a_b¡y^=Z]\e|9�Lux~�_9a1�9�y^eux�y�y~�Z]^£~�a5|9_b��|A~�a{��|A~�u��9X[~�a · �9ay�[�g�9a{�]Z]\¥z1��u`~�Z}a?u`^]Û?~�uY��_b�y���¡UZµ\=Z]zy\=Z]^=Z]a5uxZ}�Ü�9^���X[~�^`^=~�ay�æ��~�ay´+cD¡UZ}ut��Z]Z}a+|�~�u��9X[~�a · �9ay�Í�g�9ay�}Z]\xjF~�auxw{Z7�9^`^=_��}~¦��u`Z]�¸�{~¦�9�b\`�9Xµ� staæ^=�yX¤X¤�9\e�bª¬u`wyZ2zy\=~�X¤�9\=�¯�{^`Z2_9�Yu`wyZ2X[Z}u`wy_��{^�yZ}^`�]\=~�¡UZ]�æ~�a«uxwy~�^©��w1��z{uxZ}\o��~�Z]^©a{_9uo~�a«uxZ}^=ux~�ay���]�9�y^x���Æ�9^=^`�yX[z{ux~�_bay^ ¡y�{uP~�azy\=_;|A~��y~�ay�«�9aMZ4­�Z}�}u`~�|bZ��¦�9ay�9�1�9�bZF�G_b\SX��9´?~�ay�µu`wy_b^=Zv�9^`^=�yX[z{ux~�_bay^Szy\`Z}�]~�^=Z�9ay��Z��{z{��~��}~�u]�2½©^=^`�yX[z{ux~�_bay^P�]�9a'u`wyZ]\=Z]¡5��¡UZ[~�^`_b����uxZ}�«�G_b\P�yZ}��~�¡¬Z}\x��u`~�_ba¸_9\Z4��zUZ]\=~�X[Z]a5u���u`~�_baµ�9a{�µu`wyZ]aMcD_ba{�]ZP|��9��~��1��uxZ}�±j�¡UZS~�a5uxZ]�9\x��u`Z]�v��~�u`w¨^eu���u`~�^=u`~���g���°�1��u��2u`_F�A~�Z]���¯¢?�1�9a5u`~�u��Lux~�|9ZPZ]^eux~�X¤��u`Z]^�_9���g�9�y^`�9�°Z}­¬Z]�4ux^}�

½©a�~�X¤zU_b\eu��9a5u¥~�^=^`�yZruxw1��u¥��~����1¡¬Z �]_ba{^`~��{Z]\`Z}��_bay���F¡y\`~�Zi�y�[~�a[u`wy~�^¥¡¬_A_b´cD^=Z]Z,¿�Z]�4ux~�_9a×f{� ìbjv~�^¯^x��X¤zy��~�ay�,|��9\=~¦�9¡{~���~�ut�b�ÚÞ�wyZ�X���u`wyZ]X¤��u`~��g���©�yZ}\`~�|��L�ux~�_ba�_9���]�9�y^`�9��Z4­�Z}�}u�Z]^eux~�X��9a{�y^�^`w{_b�y����¡¬Zo�}_bay^=~��yZ}\`Z]�µ��®y\`^=u�^euxZ}z�ux_�����\`�^`�{zyzy��Z}X¤Z}a5ux~�a{�MuxwyZ}^`Z¸Z}^=u`~�X¤�9ay�y^���~�uxwÍ�]_ba{®y�yZ]ay�}Z�~�a5uxZ}\=|��9��^¤�9ay�Ý^=~��ba{~�®{��g��ay�]Z���Z}|9Z]��^}ªY�9^�~�a�u`\x�9�{~�ux~�_ba1�9�Æ�9a1�9����^=~�^S_9���]_9a?u`\`_b����Z}�MZ4��zUZ]\=~�X[Z]a5ux^}�¨å�Z^`w{_b�y���[\`Z}X���\`´±ª5uxwy_b�{�bw/ª5uxw1�Lu�wy��|A~�ay��_9¡{u��9~�ayZ]�[ay_ba{z1�9\x��X¤Z4ux\`~���Z]^eux~�X¤�9ay�{^�G_b\[�g�9�{^x�9�rZ}­¬Z]�4ux^��y_AZ]^¤ay_9u¤~�X[zy����uxw1�Lu�_bayZµ^`w{_b�y���ã\`Z}�G\`�9~�a¾�G\`_9X �y^`~�ay�z1�9\`�9X[Z}ux\=~��r�G_b\`X[^�~�a¤uxwyZPZ}^=ux~�X��Lux~�_9a�zyw1�9^=ZP_9��uxwyZP^eux�y���b�¥è1_b\�Z4�{�9X[zy��Z9ª{~��uxw{ZH�9^=^`�yX[z{u`~�_bay^r_9�z���9�y^=^`~��9a/ª±ä]Z}\`_��âX¤Z]�9a¨�{~�^=u`�y\`¡y�9ay�]Z}^P�9ay�«�9�y�y~�ux~�|bZ�~�a��uxZ}\x�9�4ux~�_bay^Æ��\`Z��{Z]Z]X[Z]�H\`Zg��^`_ba1��¡y��Z�ª9uxwyZ}aHuxw{Z�Z]^eux~�X¤�9ay���b~�|9Z]a2~�avcDÁ{� � f5j��]�9a¡UZ��}_ba5|bZ]\euxZ}��u`_[u`wyZHz{\`_A�y�y�}u/G�cDù¬��� j�! w )�Ï w Ð�)�� Ï �1��wyZ]\=Z w Ð�)�� Ï!~�^�uxw{Z^=ux�9ay�1��\`�y~�ä]Z]�[\`Z}�b\`Z}^`^=~�_ba[�]_AZ}¶[�]~�Z]a5uocJ¿�Z}�}ux~�_ba�ì�� Á���d;j�ÛbuxwyZrZ]^eux~�X���u`~�_baHzy\`_b¡A���Z}X�u`wyZ]aæ\=Z]�y�{�]Z]^Pu`_�uxw1��uo_9�£Z]^eux~�X¤��u`~�ay�¤\`Z]�9\`Z]^=^`~�_ba��]_AZ}¶[�]~�Z]a5ux^[cGZ9�ý�y��ª�¡?���Z]�9^=ur^=¢A�y�9\`Z}^�j4��§¨_b\=Z�^=_bzywy~�^=ux~��g��u`Z]�¯Z]^eux~�X���u`~�_ba�u`Z]��wya{~�¢?�yZ]^ �g�9a¨¡¬ZP�G_b�{ay�~�a�ê�_b^`Z}aA¡y�9�yX �9ay�Mêr�{¡y~�aãced�÷9fbÁ5j4ª�ê�_b¡y~�ay^¤ced�÷bf9÷{ª�^`Z]���£dG�bj4ª$�9ay��ê�_b¡y~�a{^

Page 45: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ�~5ÎãÒoÓeÐ�ÄPÏ�ÐyÐ�Ó4Ù�Ö dgÃo�Z}uP�9�J��ced�÷9÷ � ª�z{z/�/ÁbÁ�dQ��Á5j��Pè1_b\©Z��y��X¤zy��Z9ª�u`wyZÄà=z{\`_bzUZ]ay^=~�ut�¨^`�}_b\`ZiáµX[Z}u`wy_A�_9�oêr_b^=Z]a?¡1�9�{X �9ay�Íêr�y¡{~�a×ced�÷bf9Á5j�����^F�G_9�yay�Íu`_�¡¬Zµ¢?�y~�u`Zµ�{^`Z}�G�{����w{Z]auxw{ZS�{~�X[Z]ay^=~�_ba1����~�ut�[_9��uxw{Z����L³=�y^=u`Z]�¨�]_�|��9\`~���uxZ}^�~�^�wy~��bw/�Ysta¨�[X[_b\`ZP\=Z]�}Z]a5u^`��w{Z]X[Z'�]�9����Z]�¹à=X¤�9\=�b~�a1���rX[_A�yZ]��^]ªdá+ê�_b¡y~�a{^�ced�÷9÷b÷5j¤^=wy_���^µu`w1��ugªr\x�LuxwyZ}\uxwy�9avZ]^eux~�X¤��u`~�ay�S~�ay�{~�|A~��y�y�9�1�D�9�}u`_b\`^�~�a�uxw{ZP�9�L³=�y^euxX[Z]a5u��G_b\=X2�y���S_9��cDÁ{��dg÷5j4ª~�uH~�^�_9�QuxZ}a�X¤_b\=Z��9��|9��a?ux�9�bZ}_b�y^Hu`_æ�y^`Z��¤c������ j·!�� $����

7 »�» $1�

��� 7 « $#� ª���wyZ}\`Zvuxw{Zzy\=Z]~�a5u`Z]\=|9Z]a5ux~�_ba«�{~�^=u`\`~�¡y�{ux~�_ba¯\`Z}X��9~�ay^��ya{�D�9�4ux_b\=~�ä}Z]�/��²oayZS�g��aµu`wyZ]a¸^=Z]z1�;�\x�LuxZ]���æZ]^eux~�X���u`Z7uxwyZ[�yZ]a{_bX¤~�a1��u`_b\(��c �</5j�ª���Z]~��bwM~�ay�y~�|�~��y�1�9��^x�9X[zy��Z}^�¡?�uxw{ZH~�a5|9Z]\`^=Z2_9�Yu`wy~�^rZ]^eux~�X¤��u`Z9ª¬�9ay�«u`\`Zg�Lu u`wyZ���Z]~��bw5uxZ]�æ^`�9X[zy��Z}^©�9^ ~��Æu`wyZ}���Z]\`ZF�y\`�g��aM��u�\`�9ay�y_9X&�G\`_9X�u`wyZFz¬_b^eux~�a?u`Z]\e|bZ]a5u`~�_ba��y~�^=ux\=~�¡y��ux~�_9a ��c-�.�4� j4�´$_9^=ux~�a5uxZ]\e|bZ}a?u`~�_bavzy�9\x�9X[Z}u`Z]\=^]ªA^`�y��w¯�9^�uu

7 G�cDù¬��� j�ª��]�9a�u`wyZ]a�¡UZ©Z}^=u`~�X¤��uxZ}�¡5��_b\`�{~�a1�9\e�æ��Z]�9^=u�^=¢?�1�9\`Z}^]��Þ�w{~�^SX¤Z4uxwy_A��~�^�Z]^=z¬Z}�]~¦�������¸�9�{|��9a5u����bZ]_b�{^�~�a��_9ay�b~�u`�y�y~�a1�9�5^=u`�y�y~�Z]^Æ��~�u`w2ux~�X¤Z��â|��9\=�A~�ay� �}_�|9��\`~¦�LuxZ]^}ª?�9^Æ~�aHuxwyZ�z{\`_A�]Z]^=^Y�]_baA�ux\=_b�/zy\=_b¡y��Z]X��y~�^`�}�y^`^=Z]�«~�a¨¿�Z]�4ux~�_9a¨Á�� � �ÂÁµcD^=Z]Z�cÀÁ{��d�f5j=j4�

¿�Z}|9Z]\`�9�yZ4�Au`Z]ay^=~�_bay^Y_��yuxwyZ�X¤Z4uxwy_A�y^�z{\`_bzU_b^`Z}�7~�aHu`wy~�^$��w1�9z{u`Z]\Y��\`Z�a{_9uxZ����_b\=u`w?�9�vè�~�\`^=u]ª°uxwyZ[~��yZ}a5ux~�®y�g��u`~�_ba��9ay�9���A^`~�^P�G_b\S��ux_bX[~��H~�a5uxZ}\=|bZ}a5ux~�_9ay^H�]�9a¡UZ��bZ}ayZ]\`�9��~�ä]Z}�µu`_¤�}_bX¤z{��Z4�µzU_b��~��]~�Z]^�~�aµ��w{~���wæ�F^`Z4u ø¹_9�Æ�}_ba5ux\`_9����Z}�µ|��9\=~���9¡y��Z]^�~�^�X¤�9�yZrux_2\=Z]^`zU_bay�¯~�av�H^`zUZ]�}~�®1Z}�v���g��u`_2^`_9X¤Zo^=Z}urú×_����]_�|��9\`~���uxZ}^|A~¦�¤�G�{ay�}u`~�_ba1���$_b\ ^=u`_���w1��^=ux~��2^=u`\x��u`Z]�b~�Z]^}ª/�9^o~�aæ¿�Z}�}u`~�_ba�Á{� � �ÂÁ{�©sta · wy�9z{uxZ}\ÃvcÀ¿�Z]�4ux~�_ba�Ãy� � j4ªA~�u¥~�^�^=wy_���a�uxw1��u�~��yZ}a5ux~��Q�A~�ay�Su`wyZ©Z4­�Z}�}u�_9��^`�y��wvzU_b��~��]~�Z]^Y~�^Z]¢?�y~�|��9��Z}a?u�u`_¤�}_bX[zy�{ux~�ay�2uxw{ZSZ���zy\`Z}^`^`~�_baÇ�¤c������ �:/5j��

½�^=Z]�]_9ay�ÜZ4�AuxZ}ay^`~�_ba+�]_9ay�]Z}\`ay^�u`wyZ«�y^=Z¸_9�Pu`wyZ«~�a?u`Z]\e|bZ]a5u`~�_ba+�]�9���}�y���{^cDÞ�w{Z]_b\=Z]X�Á{�ÂÃ{��d;jY~�a[ay_ba{\`Z]�}�y\`^=~�|9ZoX¤_A�yZ}��^]ª?uxwy��u�~�^]ªA~�a[�g�9�{^x�9�U�y~¦���b\x�9X[^£~�a��|b_9��|A~�ay�P�{~�\`Z}�}u`Z]�[�}�A�]��Z]^£_b\Æ�GZ]Z}�y¡1�9��´7��_�_9zy^]��Þ�wyZr¡1�9^`~����yZ}®yay~�u`~�_ba7_9�U�]�9�y^x���Z}­¬Z]�4ux^7~�a�uxZ}\`X _9�Ýàe��~�zy~�a{�«_b�{uZá¸Z]¢?�1�Lux~�_9ay^H�G\`_9X�uxwyZ�X[_A�yZ]�©c4²oZ4®1ay~�u`~�_baÁ{� � ��d;j¥^eux~����±�g�9\=\`~�Z}^�_�|bZ}\�ux_2a{_bay\`Z}�]�y\=^`~�|bZP^=�A^euxZ]X[^�cÀ¿Aux\=_9uxäP��ay��å�_b���¸dg÷bhbi{Û¿�_b¡UZ]��d�÷9÷bi5j4ª?¡y��u�u`wyZ]a[ut��_�~�^`^=�yZ]^�X2�y^eu�¡UZ©���y�y\`Z}^`^=Z]�/�¥è�~�\`^=u]ªbuxwyZ©�9a1�9���A^`~�^_9�£~��yZ]a5ux~�®1�g�Lux~�_9a¸X2�y^euPZ]ay^=�y\`Z2u`wyZ7^=ux�9¡y~���~�ut�¯_��YuxwyZ7\=Z]X¤�9~�ay~�ay�[^`�y¡yX[_A�yZ]��^cÀè$~�^`w{Z]\vd�÷o��i5j4�¾¿AZ]�]_9ay�/ª£uxwyZ¬ê9�t^=Z]z1�9\`��ux~�_ba��]\=~�u`Z]\`~�_ba,�G_b\�²�½1��^7XH�y^=u[¡UZZ4�Au`Z]ay�yZ}��ux_¸�}_;|9Z]\7�4���}��~����b\`�9zywy^H�9^H��Z]���À�¨Þ�w{Z�|��9��~��y~�ut��_9�mêb�â^`Z}z1�9\x�Lux~�_9aw1�9^�¡¬Z}Z]a�Z}^=u���¡y��~�^`wyZ}�M�G_b\Ha{_bay\`Z}�]�y\=^`~�|bZ���~�ayZ]�9\�X[_A�yZ]��^�cÀ¿�zy~�\euxZ}^�d�÷b÷bìbjS��^��Z]���©�9^¤�G_9\�ay_ba{��~�a{Zg�9\[^=�A^=u`Z]X[^v~�a5|b_9��|A~�ay���y~�^=�]\=Z}uxZ¨|9��\`~¦��¡y��Z}^¸c�´$Zg��\`�o��ay�²oZ]��w5u`Z]\¯d�÷b÷bhbj4�Ó³ _���Z}|bZ}\]ª¥uxwyZv�}_bX¤z{�{u���u`~�_baM_9�©�]�9�y^`�9��Z}­¬Z]�4uFZ]^eux~�X¤�9ay�{^��~����/¡UZ�w1��\`�yZ}\ ~�aµ�4�A�]��~���a{_bay��~�ayZg��\�^=�A^=u`Z]X[^]ªU¡¬Z}�g�9�y^=ZH^e��XH¡¬_b��~��P\=Z]�y�y�4ux~�_ba_9�Â��c-�.�4� j�ux_Fw1�Lu=�â�G\=Z]ZPZ��{z{\`Z]^=^`~�_bay^©X¤�g��\=Z]¢?�y~�\=ZPuxwyZP^=_b���{u`~�_bav_���ay_9ay��~�ayZg�9\Z]¢?�1�Lux~�_9ay^]�Ysta · w1��z{uxZ}\1�vcJ¿�Z}�}ux~�_ba���� � ��d;j¥��Zo�yZ}X¤_ba{^=ux\`��uxZ uxwyZPZ4|9�����1��u`~�_ba

Page 46: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

dgÃ5fyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê_9��z¬_b��~��}~�Z]^��9ay�«�}_b�ya5uxZ}\=�D�9�4ux�1����^�~�aµay_bay\=Z]�}�y\`^=~�|bZ���~�ayZg�9\�^e�A^=uxZ}X¤^HcD^=Z]ZH�9��^=_ã��9��´9ZS�9ay�¬´�Zg�9\=��cOd�÷b÷5ì9j`j4�

½ uxwy~�\`�¸Z��AuxZ]a{^`~�_9a'�}_bay�}Z]\`a{^S�9Z]ayZ}\x�9��~�äg�Lux~�_9ay^ _���~�a5uxZ]\e|bZ}a?u`~�_baæ�]�9���}�y���{^cDÞ�w{Z]_b\=Z]X Á{�ýÃy��d;j�ux_F^`~�ux�1�Lux~�_9ay^���wyZ]\=ZSu`wyZ��1��ux�[�g|��9~��¦�9¡y��Zo~�^�a{_9u�_b¡{ux�9~�ayZ}��yay�{Z]\F~J�ý~J�ý�/��cG~�ay�{Z]zUZ]ay�yZ}a5u��9a{�¼~��yZ]a5u`~��g���������y~�^=ux\=~�¡y��uxZ]�1jH^x�9X[zy��~�ay�y�,²Pa{Z�g��aM~�X¤�9�b~�ayZ9ª¬�G_b\�~�ay^eu��9ay�}Z9ª$�µz{w?�A^=~��]~��9a���wy_«zy\=Z]^=�]\`~�¡¬Z}^2�¯�]Z}\=u���~�a�ux\`Z]��u=�X[Z]a5u�u`_¸z1��u`~�Z}a?u`^�_9ay���æ��wyZ}a,uxwyZ[�G\x�9�4ux~�_ba�_���^`�y\e|A~�|b_9\`^2��X¤_ba{�¨zy\=Z}|A~�_9�y^z1��u`~�Z}a?u`^°�y\`_9zy^�¡¬Z}��_��M^=_bX[ZÆuxwy\=Z]^`w{_b���/��staS^`�y��wS�g�9^=Z]^}ª9~�u/~�^°\`Z}¢?�y~�\=Z]��ux_�Z]^eux~��X¤��uxZ�u`wyZ �]�9�y^`�9�±Z}­¬Z]�}u��¤c������ j��G\=_bX ay_baA�t~�a{�yZ]zUZ]ay�{Z]a5u�^`�9X[zy��Z}^]�eµ©�¦�9�{~�X[~�\µ£_�|A´Ýcedg÷b÷bh5j��5�g|bZv�]_9ay�y~�u`~�_ba{^��ya{�yZ]\2��wy~���w�uxwyZ�\=�y��Z}^2_9��Þ�wyZ}_b\`Z}X Á{�ýÃy��d��~�����¡UZF�9zyzy��~��]�9¡y��ZS��wyZ]a'^x��X¤zy��~�ay��~�^oay_�u�~J�ý~J�ý�/��ª/�9ay��wyZ2��Z]a5uH_9a�u`_v�g�9^euuxw{Z�u`wy\`Z}Z�~�a{�GZ}\`Z]a{�]Z�\`�y��Z]^r�9^r�7��_b�9~��g����zy\=_��y�{�}ux~�_bav^=�A^euxZ]Xµ�

�m���m��� ö¶°'�·y��>�0w ¡¤��¡¤�é£ �®> [§�w �®>°¹~V��¨�����­?ynx[�·y�§Þ�wyZ¸¡UZ]a{Z}®yuv_��P~�ay�}_b\`zU_b\`��ux~�ay��^`�y¡y^eu��9a5u`~�|bZæ¡y�9��´?�b\`_b�{ay�Ý´?ay_�����Z]�{�bZ¸~�a5u`_zy\=_b¡1�9¡y~���~�^=ux~��¨~�a{�GZ]\=Z]ay�}Zæ���9^¯\=Z]�]_9�bay~�ä}Z]�á�9^��D�9\v¡1����´Ý�9^vÞ�wy_bX¤�9^�ã��g�9Z]^ced2�9hbÁ5j/�9a{� ´Y~�Z}\`\`Z����9zy���9�]Z cedgf{dgÃ?j�ª���ay��~�ux^��]\=�y�]~��9�?\`_b��ZY~�a�uxwyZ��9a1�����A^`~�^$��ay�~�a5u`Z]\`z{\`Z}ux��ux~�_ba«_��£�]_bX[zy��Z��¨^=u��Lux~�^eux~��g�9��^=u`�y�y~�Z]^o~�^©�bZ}ayZ]\`�9�����µ�9��´?ay_�����Z]�{�bZ]�¡5�ãX[_b^eu�X[_��yZ}\`a+^eu���u`~�^eux~��}~¦�9a{^]�¶³r_���Z4|bZ]\}ª©uxw{Z¸X��LuxwyZ}X���u`~��]�9���¦�9a{�b�1�9�bZ�g|��9~����9¡y��ZÆ�G_9\�Z4��zy\=Z]^=^`~�a{�P¡1�9��´?�b\`_9�yay�7´?ay_�����Z}�y�bZ�w1��^�\`Z}X���~�ayZ}�H~�aH�©\x�LuxwyZ}\zy~�ux~��G�{��^=ux��uxZ�_9�$�yZ4|bZ]��_bzyX[Z]a5ug�

Þ$\`�9�y~�ux~�_9a1�9�����bªg^eu���u`~�^eux~��}~¦�9a{^/w1�g|bZ��9zyzy\=_;|9Z]��_9�{_9ay���P_9ayZY���g��_9���]_bXH¡y~�a��~�a{��^`�{¡y^=ux�9a5ux~�|9ZF´Aa{_;����Z]�y�9ZH��~�u`w¸^=ux��ux~�^=ux~��g�9���1�Lu��{ÿ�uxw{Z·ã��g�bZ}^`~¦��a'X[Z}u`wy_A�_9�/�9^`^=~��bay~�ay�S^`�y¡A³=Z}�}u`~�|bZ©zy\`~�_b\`^Yux_��y~�^eux\=~�¡y�{u`~�_bay�9�{z1��\x�9X[Z}u`Z]\`^}�YÞ$_S~�a{�]_b\=z¬_��\x�LuxZ©�]�9�y^`�9��~�a{�G_b\`X¤��u`~�_ba7��~�u`wy~�a[uxwy~�^��G\`�9X¤Z4��_9\`´±ª{zy���9~�a[�g���y^x�9�¬^=ux��uxZ}X¤Z}a?u`^^`�{��wá��^Üà`ù ~�^�ay_�u¯��­¬Z]�}u`Z]�Ý¡5�¼ø á,X2�y^euv¡UZ¸�]_ba5|9Z]\=u`Z]�Ü~�a5u`_,^`Z}a?u`Z]ay�}Z]^_b\�Z4|bZ]a5u`^ �]�9z1�9¡y��ZP_9��\`Z}�]Z]~�|A~�ay�Fzy\=_b¡1�9¡y~���~�ut�[|��9���yZ]^ScGZ9�ý�y�1�]_9�ya5uxZ]\e�D�9�}u`�1�9��^�j4�è1_b\�~�ay^eu��9a{�]Z9ª1u`_¤�}_bX¤XH�yay~��]��uxZou`wyZS~�ayay_A�]Z}a5u©��^`^`�{X¤z{u`~�_ba¯uxwy��urX2�y�µ�y_�Z}^ay_9u¥�]�9�y^`Zr\x��~�a/ªb��Zr��_b�y���¤wy��|9Z u`_S�{^`Z �S\`��uxwyZ}\£�yaya1��u`�y\x���1Z4��zy\`Z}^`^=~�_bav��ay�^x�g�[uxw1�Lu�uxwyZ©zy\`_b¡y�9¡y~���~�ut�7_��°uxwyZo�}_b�ya5uxZ}\=�D�9�4ux�1���/Z}|9Z]a5u à=\x��~�a�~���~�u¥��Z}\`ZPay_�uX2�y�{�{��á�~�^�uxwyZ�^x�9X[Z2��^ uxw{ZHzy\=_b¡1�9¡y~���~�ut��_9�,à=\`�9~�aµ~���~�ur��Z}\`Z2XH�y�y�{�9�\á+staA��yZ}Z]�/ª9u`wy~�^�~�^�wy_���u`wyZ�zU_9u`Z]a5ux~��9���â_b�{ux�}_bX[Z£�9zyzy\=_5�9��w2_9�yé Z4��X¤�9aH�9ay�7ê��y¡y~�aw1�9^o�9��wy~�Z4|bZ}�'^eu���u`~�^eux~��]�9����Z]�9~�ux~�X����}�Uÿ��g���y^x�9�1³=�y�y�bX[Z]a5u`^P�9\`ZHZ4��zy\`Z}^`^=Z]�,��^�]_9ay^=u`\x�9~�a?u`^o_baæzy\=_b¡1�9¡{~���~�ut�v�G�yay�4ux~�_9ay^o~�a5|9_b��|A~�a{���]_9�ya5uxZ]\e�D�9�}u`�1�9��|��9\=~¦�9¡{��Z]^cD^=Z]ZH¿�Z]�4ux~�_baµÁ{� h�� Á5j��· �9�y^`�9���y~��9�b\`�9X¤^�_9­¬Z]\H�9a��9��uxZ}\`a1��u`~�|9Z¤���9ay�b�1���bZ[�G_9\��}_bX2¡y~�ay~�ay�µ�1��ux�

Page 47: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ�~5ÎãÒoÓeÐ�ÄPÏ�ÐyÐ�Ó4Ù�Ö dgÃb÷��~�u`wM�]�9�y^x����~�a{�G_b\`X¤��u`~�_ba/�[Þ�wy~�^H�¦��ay�b�1�9�9Z[^=~�X[zy��~�®1Z]^ou`wyZ¸ã����9Z]^=~¦�9a,\=_b�{u`Z¡5�2���]�]Z}z{ux~�ay�Pzy���9~�a��g�9�y^`�9�{^eu���u`Z]X[Z]a5ux^Æ�9^Æ~�u`^�¡1��^`~���zy\`~�X¤~�ux~�|bZ]^}�$¿��y��wF^=ux��uxZ��X[Z]a5ux^}ª?��w{~���w�X[Z]\=Z]���2~�a{�y~��]��uxZ���wyZ4uxwyZ}\����g���y^x�9�1�}_bayayZ}�}u`~�_ba[¡¬Z4ut��Z}Z]a�ut��_|��9\`~��9¡y��Z]^2_9� ~�a5u`Z]\`Z}^=u[Z4��~�^=ux^}ª��9\=Zv�]_bX[X[_bay�����y^=Z]�ã~�a�_b\`�{~�a1�9\e�,�y~�^`�}_b�y\`^=Z�9ay�Ýz{\`_�|A~��yZæ�,ay��ux�y\`�9� ���g�ã�G_b\�^`�]~�Z]a5ux~�^=u`^�ux_��}_bX¤XH�yay~��]��uxZ¨Z4��zUZ]\`~�Z]ay�}Z�9ay�ã_b\`�5��ay~�ä}Z¨´?ay_�����Z}�y�bZ��W� sâu��]�9aÝ¡UZ¸�9a5ux~��]~�z1��uxZ}�/ª�u`wyZ]\=Z}�G_b\=Z9ª�uxw1�Lu�uxw{Z�¦��ay�b�1�9�9Z2_9�¥�g�9�{^x�9�Y�b\`�9zywy^o��~�����®1ay���9zyzy��~��]��ux~�_bay^©~�aæzy\`_9¡y��Z}X¤^o\=Z]¢?�y~�\`~�a{�^`�{¡y^=ux�9a5ux~¦���°�y_bX¤�9~�av´?ay_�����Z}�y�bZ��

Þ�wyZo���9ay�b�1���bZ�~�^¥ay_9u�a{Z}���YÞ�wyZ �y^`Zo_��°�y~¦���b\x�9X[^¥�9ay�¤^=ux\=�y�}u`�y\x����Z]¢?�1�;�ux~�_bay^�X¤_A�yZ]��^Su`_¸�]_9a?|9Z}���]�9�y^`�9��~�a{�G_b\`X¤��u`~�_ba'w1�9^2¡UZ]Z}a�¢?�y~�u`Z�zU_bzy�y���9\H~�auxw{ZS^=_��}~¦�9�°^=�]~�Z]ay�}Z]^ �9ay�«Z}�]_ba{_bX¤Z4ux\=~��]^}��¿?u���u`~�^=u`~��}~¦�9ay^}ª{wy_���Z}|9Z]\]ª¬w1�g|bZS�bZ]aA�Z]\`�9�����ã�G_b�{ay�!u`wyZ]^=Z�X[_A�yZ]��^v^`�y^=z¬Z}�}ugª�zUZ]\=w1�9zy^µ¡¬Z}�g�9�{^`Z'^`_A�]~¦���o^`�}~�Z]a5u`~�^=u`^�9ay�«Z]�]_9ay_bX[Z}ux\=~��}~¦�9ay^�w1�g|9ZH�D�9~���Z]�¯ux_¤zy\`_�|A~��yZH�9a«�{a1�9X2¡y~��b�y_b�{^ �yZ4®1ay~�ux~�_9a_9�Pu`wyZ«Z}X¤zy~�\`~��g�9���}_ba5uxZ]a5uv_��PuxwyZ}~�\[X¤_A�yZ]��^O»+uxw1��u[~�^]ª�ux_M^`zUZ]�]~��Q�¾uxwyZ¸Z��?�zUZ]\`~�X¤Z}a5u��9�r�]_ba{�y~�u`~�_bay^}ª�wy_���Z4|bZ]\¯w5�Az¬_�uxwyZ4ux~��]�9�Jª���wy_9^`Z�_b�{u`�]_bX[Z]^���_b�y���¡UZµ�}_bay^eux\x��~�ayZ}�Í¡5�¼���b~�|9Z]a¼^=ux\=�y�}u`�y\x����Z]¢?�1��u`~�_ba/�×c · wy�9z{uxZ}\¤ì��y~�^`�}�y^`^=Z]^uxw{Z¨¡y~�äg�9\=\`Z¯wy~�^=ux_9\=��_9�©^=u`\`�y�4ux�y\`�9��Z]¢?�1��u`~�_bay^[~�a�u`wyZ¨^=_��}~¦�9��^=�]~�Z]ay�}Z]^���ay�Z]�}_bay_bX[~��}^�j��¨½ ^7�«\`Z}^`�y��ugª$Z}|bZ}a¾^`�y��w¾¡1�9^=~��[ay_9u`~�_bay^H�9^ àe^=ux\=�y�}u`�y\x�����}_�Z4�¦�®1�}~�Z]a5u`^ZáM_9\ÈàeX¤~�^`^`~�ay�æ��~�ay´?^Zá'¡¬Z}�]_bX[ZvuxwyZµ_b¡A³=Z}�}u¤_9�o^=Z]\=~�_b�y^F�}_ba5ux\`_�|9Z]\`^e�cÀè1\=Z]Z]�{X��9a�d�÷bfo�AÛE�P_b���{¡¬Z}\`�bZ}\Sdg÷b÷ � j��9ay�¸X[~�^=~�a5uxZ}\`zy\=Z}ux��ux~�_bay^�cGåÜwy~�u`ux�9´9Z}\d�÷b÷9i{ªyz/�±Ábi � Û�å�Z]\=X2�{uxw'd�÷b÷ � Û · _g�¨�9ay�vå�Z]\`XH�{uxw'd�÷b÷bÁbj4�Þ$_¸�«�¦�9\=�bZ[Z4�AuxZ}a?u]ªÆuxwy~�^Hwy~�^=u`_b\=�'_9���}_ba5ux\=_;|9Z]\=^=�¾�9a{�,X¤~�^`�}_bX¤XH�yay~��]�L�

ux~�_bav^=u`Z]X[^��G\`_bXÚuxw{Z��9¡y^`Z}ay�]Z�_9���9a¯�9�{Z]¢?�1��u`ZSX¤��u`wyZ]X¤��ux~��g�9�±ay_�u���u`~�_ba��G_9\�yZ4®1ay~�a{��¡1��^`~���ay_�ux~�_9ay^£_9���]�9�y^`�9�±X¤_A�yZ]��~�ay�{��è1_b\YZ��y��X¤zy��Z9ª?^=ux�9ay�1��\`�[zy\`_b¡A��9¡y~���~�^=ux~���a{_9u���u`~�_baH�g�9aya{_9u£Z4��zy\=Z]^`^£uxw{Z�Z]X[zy~�\=~��g�����}_ba5uxZ]a5u¥_9�±uxw{Z��]_AZ}¶[�]~�Z]a5uæ©~�a¯u`wyZ�^=u`\`�y�4ux�y\`�9��Z]¢?�1��u`~�_ba���!Ìæ �¢¾ZУª1Z}|9Z]a¨~��$_bayZ�~�^�zy\=Z]z1��\`Z]�µux_[�9^O�^`�{X¤Z2u`w1��u?¾ZкcÀ�9a��yay_9¡y^`Z}\=|bZ}�M¢?�1�9a5u`~�ut�{j©~�^P�ya{�]_b\=\`Z]����uxZ}�æ��~�uxw¸ø,���¼é _9\�g��a��9a5�'zy\`_b¡y�9¡y~���~�^eux~��HX¤Z]�9ay~�a{�¨¡UZ���u=u��9��w{Z]�Mu`_¨uxw{Z��9a1�����A^=u]ûÂ^SZ4���]���y�y~�ay�� üVFk}mK�a¿P:K�v_y\�i~2}�K�H�ma4FZK�Y2F3a�W^I�l�J4DGL)WVfgD_K�`GJ4Fka�Q8L\H_fky\X5Y2L\H_h�J-U|I�a4FklSFka�FkF3W^I�l�J4DGL)WEv;I0I*PGTf'y)K�W�W4Ldó_FZYjJ4DGF�}�FkJ4DGI2YGW.`Ga4FZW�F3H0J4F3YõDGF3a4F�KiWVv;FkyMI�HGhiL\HGh�J�ImJ4D_FgS��EK���FZW�L)K�HõfkK�}�`ÂUmK�H_YOKiWY2F3`5F3H_Y2LMHGh[I�H,K¢S4h�II2YO`Ga�L\Iia3{ U CEDGL)WÂfkyMKiW4W4L\ó5fkK�J4LMI�H?L)WÂ}�LMW4y\FZK�Y2LMHGh5{�CEDGF�}�F'J4D_IYèY2IF3WY2F3`5F3H_YÑI�H W4XGv��-FZf�J�L\xiF[KiW4W4XG}�`2J�L\IiH_W[QSF�{ h_{M~�}�X_YÑY2IF3W�HGI�JAf3K�X_W4FjagK�LMH5T'~�vGX2J1W4X_fgDÑK�W�úW4XG}�`2J4LMI�H_W|K�a�F�fkK�X_W�K�yq~iHGI�J^W�J�K:J�LMW�J4L)fkK�y�~K�H5YAfkK�HGHGI�J^v5FeF'�2`Ga�F3W�W�FZY�KiWw`Ga�LMI�a�`_a4Iiv_K�vGLMyMLdJ�L\FZWI�Hõ`_K�a�K�}�FkJ4F3a�W|I�l3�-I�LMH0J�Y2LMW�J4a�LMvGX2J4LMI�H5Wk{�#�>I�yMXG}�L\H_I�X_W|yMLdJ�FkagK:J4X_a4FeI�HAJ�DGF�W�XGvI�-F3f�JEI�l�S4F'�2I�hiFkHGF3LdJ-��UAQ8F�{ h_{üVL)fgD_K�agYOb3ñ*�QR_c � HGh�yMFF'J.K�y�{�bZñQ���_c���FkH5Y2a4�?b3ñ�ñ0ò�T�D_K�W|F3}�F3a4hiF3Y1lSa4Ii}ÜF3fkI�HGIi}�LMW�J�W1�2W-J�a4XGhih�yMF.J�I�h�LMx�F�W-JgK:J�LMW�J4L)fkK�yLMHiJ�Fka�`Ga�F'J�K�J4LMI�H[J�IAJ4D_F�f3K�X_W�K�y�K�W�W�F3a�J�L\IiH�S�� K�H5YA�|��K�a�F�XGH5f'I�a�a�Fky)K:J4FZYIU(Q���y)Y2a4L)fgD�bZñ�ñ��_cW4FkF��F3f'J4LMI�H(ò2{ �_{ �iT'{

Page 48: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d;ì9iyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê�G\`_9X uxwyZrZ]¢?�1��u`~�_baF|��9\`~��9¡y��Z]^Yuxw1�Lu���\`Zrwy~��bw{���7�}_b\`\=Z]�¦�LuxZ]�F��~�u`w¤ø&_b\�ù¡y��u�y_Fay_9uÑà=�{~�\`Z}�}u`���v�L­�Z}�}u�á[ù��N�

Þ�wyZ�ay_9ux��ux~�_ba��yZ4|bZ]��_bzUZ]�¯~�a�u`wy~�^���wy�9z{uxZ}\��b~�|9Z]^�u`wyZ]^=Z�cG�g�9�{^x�9�¦j�ay_9u`~�_bay^�F�]��Z]�9\rZ]X[zy~�\=~��g���/~�a5uxZ}\`zy\=Z}ux��ux~�_ba/ª1¡UZ]�g���y^`Z�~�urz¬Z}\`X[~�u`^�_bayZ�ux_�^=z¬Z}�]~��Q�¯zy\`Z���]~�^`Z}���ã��wy��uµ~�^v¡UZ]~�a{��wyZ]���á�}_bay^=ux�9a5u¨�9ay�Ý��w1��uµ~�^vX[Z]\=Z]���+X[Zg�9^=�y\`Z}�á~�a�ã�b~�|bZ}aþZ4��z¬Z}\`~�X¤Z}a?u]� cDÞ�wyZ�a{Z]Z]�þ�G_b\µuxwy~�^«�y~�^=u`~�ay�4ux~�_9aá���9^æ\`Z}�]_b�ba{~�ä]Z}�¡5�æX¤�9a5�æ\`Z}^`Zg��\`��wyZ}\`^]ª$X¤_9^=uPay_�u��9¡y��� ´Y\x�Lu`uS�9ay�M¿���wy���9~��GZ}\2d�÷bf9fµ��ay� · _��d�÷b÷ � j4��Þ�wyZ�X¤Z]�9ay~�ay�o_9�^æ�~�^Y^`~�X¤zy��� uu

7 G�cÀù¬��� j4ªbu`w1��uY~�^]ª9u`wyZ�\x�LuxZ�_9�U��w1�9a{�bZcD~�a j©_9�Yu`wyZ7Z4��zUZ]�}ux��ux~�_baæ_9�¥ù�~�aæ�9aæZ4��zUZ]\=~�X[Z]a5uo��wyZ}\`Z7ø ~�^©wyZ]������u ¡5��Z���u`Z]\=a1�9�/�}_ba5ux\`_9�J�ÆÞ�wy~�^�~�a5uxZ}\`zy\=Z}ux��ux~�_ba�wy_b���y^�\=Z]�5��\`�y��Z]^`^�_��°��wyZ}u`wyZ]\1¾ZÐ�9ay�æø �9\=Z[�}_b\`\=Z]����uxZ}�¼cDZ9�ý�y��ª°|�~��¯�9ay_9u`wyZ]\PZ}¢A�y��ux~�_ba !é½o�p� ¾kÏ�j4� ��~�´9Z����~�^=Z9ª{u`wyZP�9ay�9���A^=u]ûÂ^��yZ]�}~�^`~�_ba¯�9^�u`_2��wy~���w�|��9\=~¦�9¡{��Z]^�^`wy_9�y���v¡UZo~�a{�]���{�yZ]�v~�a�P�b~�|9Z]a[Z]¢?�1��u`~�_ba[�g��a¤¡UZ�¡1�9^=Z]��_9a��Pw5�Az¬_9u`wyZ}u`~��]�9�1�]_ba5u`\`_b����Z]�FZ��{zUZ]\=~�X[Z]a5ugÿ½*|��9\=~¦�9¡{��Z�ú~�^ Z��{�}���y�{Z]�¸�G\`_9X u`wyZHZ}¢?�1��ux~�_ba¨�G_9\©ù ~��rcQ�G_b\ Z4|bZ}\=�¨��Z}|9Z]��_9�¾ZÐ$jPú w1�9^�ay_¯~�a��1�yZ]a{�]ZF_ba'ù���wyZ]a��9���$_9uxw{Z]\o|��9\`~��9¡y��Z]^Fc¿�|Ð�)¬j4ª��9\=Z7wyZ]����]_9ay^=ux�9a5ugÛ£uxwy~�^�~�X¤zy��~�Z}^,�¤c������/��s��Ð�)�j,! �¤c����-��ÐC)¬j4�¸¿�zUZ]�]~�®1�g�������9ª�|��9\=~¦�9¡{��Z]^uxwy��u �9\`Z�Z4���}���y�yZ}�¨�G\=_bX�uxw{Z�Z}¢A�y��ux~�_baâ�·!¶æ � ¾ZÐ���\`Z�ay_9ur�]_9ay�y~�u`~�_bay�9�����~�a{�yZ]zUZ]ay�{Z]a5u[_9�où��9~�|bZ}a¾X¤Z]�9^`�y\=Z]X[Z]a5ux^F_9��ø ¡y�{uF~�ay^=u`Zg�9�¼�9\=ZM@`>9ç?:]>9CQCE3~�\=\`Z}��Z}|��9a5uSux_�ù��b~�|9Z]a�^`Z4u`u`~�ay�b^H_9��ø��¸Þ�wyZ�_bz¬Z}\x��u`~�_bay�9�£X¤Z]�9ay~�a{�«_9��uxw{Zà=�y~�^=u`�y\`¡1��ay�]ZPuxZ}\`XÝáݾZÐ,~�^���~�´�Z}��~�^=Z��yZ]XH�A^eux~�®yZ]�/ÿ�¾ZÐ,~�^��yZ}®1a{Z]�¸��^�u`wyZ��y~��¦��GZ]\=Z]ay�}Z�ù H;G�cÀù¬����Ð�j4�$Þ���_��y~�^=ux�{\`¡1�9a{�]Z�u`Z]\`X[^]ª�¾3Ͼ�9ay� ¾ZÐ�ªb�9\`Z��]_b\=\`Z]����uxZ}�~��V��c-�.��� �s��0Ï�Ð�j m!��¤c���� �o��ÏÂÐ�j4ª��9ay��^=_7_ba�cG^`Z]Z · w1�9z{u`Z]\�ì�ª{¿�Z}�}ux~�_ba¯ì��ýÃ��G_9\�G�y\euxwyZ}\rZ}�¦�9¡U_b\`��ux~�_ba±j4�

Þ�wyZH�{~�^=u`~�ay�4ux~�_bay^�zy\=_;|A~��yZ]�¨¡?�vu`wyZSw1��uray_9ux��ux~�_ba¯�]�¦��\`~��Q�¤uxwyZSZ]X[zy~�\`~��]�9�¡1�9^=~�^r_9�Y^eux\`�{�}ux�{\x�9��Z]¢?�1��u`~�_ba{^o�9ay�«^`wy_b�{���«X¤�9´�Z��]�9�y^x���$X[_��{Z]��^rX[_b\`ZS�9����]Z}z{u���¡y��ZÆu`_rZ}X¤z{~�\`~��g�9�L\`Z}^`Z]�9\`��wyZ}\`^}�£§¨_b\=Z]_�|bZ}\]ª9^=~�ay�}ZYX¤_9^=u�^`�}~�Z}a?u`~�®1�Y´?ay_���� �Z]�{�bZ�~�^�_9\`�5�9a{~�ä]Z}�2�9\=_b�yay�SuxwyZ¥_bzUZ]\`��ux~�_ba�_���à=wy_9���y~�ay��ø�®{��Z]��áo\x��u`wyZ]\�uxwy�9aà=�}_bay�y~�ux~�_9ay~�ay�P_9a[ø�ªdá�uxwyZray_9ux��ux~�_ba[�9ay�[�g�9���]�y���y^Y�yZ}|9Z]��_bz¬Z}��~�a7u`wy~�^£��w1�9zA�uxZ}\¯^`wy_b�{���Üzy\`_�|A~��{Z���aáZ4­�Z}�}ux~�|bZ'X¤Z]�9ay^��G_b\v^=�]~�Z]a5ux~�^=ux^¯ux_¾�}_bX¤XH�yay~��]��uxZ^`�{¡y^=ux�9a5ux~�|9Z�~�a{�G_b\=X���u`~�_ba��9ay�vux_[~�a{�GZ}\�~�u`^���_b�9~��g�����]_9ay^`Z}¢A�{Z]ay�}Z]^]�

�3CEDGFèvGLdJ4J4F3aõfkI�H0J4a�I:x�F3a�W4��v;F'J-U|FkFkH�ôÂI�y)Y2v5F3a4hiFkaèQ-b3ñiñi÷iTmK�H_Y�W,F3a4}�X2J4DÓQ�b3ñiñi÷�Tma4Fkúx�Iiy\xiF3WAK�a4IiXGH_Y£WèFka�}zX2J�D�� WALMH_W�L)W�J4FkH5f'F?IiH h�LMxL\HGh K�W-JgK:J�LMW�J4L)fkK�yeLMH0J4Fka�`Ga�F'JgK:J4LMI�HÑJ�I�J4D_FökF3a4I�f'IF#�mfkL\F3H0J�WVLMHOW-J�a4X_f'J4XGagK�y�FZ�iX5K:J4LMI�H5W�Q8W4FkFz�FZf�J4LMI�HOò2{ �_{MbZT'{

Page 49: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ�~5ÎãÒoÓeÐ�ÄPÏ�ÐyÐ�Ó4Ù�Ö d;ìAd�m���m�g� ñ �>��­[¡�¨k�®�°k¥�­ ª��>¥Dw ���>���O [¡é®�¥ &�¥�®�§.­�®�°k��¨l �xj®�õ~V¥Dw §�¡Þ�wy~�^���w1�9z�uxZ]\¤�y^`Z}^�ut��_,\`Z}zy\`Z}^`Z]a5ux��ux~�_bay^�_9�o�g�9�y^`�9�r~�a{�G_b\=X��Lux~�_9a/ÿ¨�9\x�9zyw{^�9ay�Ü^eux\`�{�}ux�{\x�9�oZ}¢?�1��ux~�_bay^}ª ��wyZ}\`ZæuxwyZ¸�G_9\`X[Z]\�~�^¯�9aÜ�9¡y^eux\`�9�}u`~�_baÝ_9�Suxw{Z�¦�Lu`uxZ}\]�Oã�_9uxwæ\=Z]zy\=Z]^`Z}a5u���u`~�_bay^Pwy��|9Z7¡¬Z}Z]aæ�}_ba5ux\`_�|9Z]\`^=~¦�9�$�G_b\o�9��X¤_9^=u©���]Z]aA�ux�{\=�b�¥²oa�u`wyZo_ba{Z©w1�9a{�/ª{Z}�]_bay_9X¤~�^=ux^��9ay�v^=_��}~¦�9�±^`�}~�Z}a?u`~�^eux^�w1�g|9ZoZ]X2¡{\x�9�}Z]�uxw{Z]^`ZæX[_A�yZ]��~�ay�æux_A_b��^]ª�¡y�{u�uxwyZ4�ã�}_ba5ux~�a?�yZ¨ux_,¢?�yZ}^=ux~�_baÜ�9ay�ã�yZ]¡1�LuxZ¸uxw{Z�g���y^x�9���]_ba5u`Z]a5u7_9��u`wyZ�z1�9\`�9X[Z}uxZ}\`^SuxwyZ4�MZ}^=ux~�X��LuxZ«cD^`Z}Zµ¿�Z]�4ux~�_bay^7ìA��dv��ay�ì��ýÃ��G_b\v�yZ}ux�9~���^�j�Û���^¨�¾\`Z}^`�y��ugªruxwyZ��y^=Z�_9��^=u`\`�y�4ux�y\`�9�PX[_��yZ}��^v~�a!zU_b��~��}�5�X¤�9´?~�ay�µ�]_ba5u`Z4�Aux^2~�^�_��QuxZ]a'|A~�Z}��Z]�,��~�uxw'^`�y^=zy~��}~�_ba��«¿?u���u`~�^=u`~��}~¦�9ay^}ª�_baMuxw{Z_9u`wyZ]\�w1�9a{�/ª�\=ZO³=Z}�}u�¡¬_�uxw�\`Z}zy\`Z}^`Z]a5ux��ux~�_bay^$�9^°zy\`_9¡y��Z}X���u`~���cDè1\`Z]Z}�yX¤�9aFd�÷bfo�bj~���ay_9u¥X[Zg�9ay~�ay�b��Z]^`^ cQåMZ}\`X2�{u`w«d�÷9÷ � Û�³ _b���¦�9ay�¯d�÷9÷5ìbj4ª?��ay�¤u`wyZ}�[^`_9X¤Z4ux~�X[Z]^\`Z}^`_b\eu�ux_7uxwyZSérZ}�AX���a��Oê��y¡y~�aµzU_9u`Z]a5ux~��9���â_b�{ux�}_bX[Z©ay_9ux��ux~�_ba���wyZ}a«zy\=Z]^`^=Z]�ux_M�]_bX[X2�yay~��g��u`Z«�]�9�y^x���o~�a{�G_9\`X¤��ux~�_baácÀê��y¡y~�a×d�÷b÷bi5j�� �Dü ½��yZ4u��9~���Z}�ã�G_b\e�X¤�9�/�9ay�9���A^`~�^�_9��uxwyZ�\`Z}�¦��u`~�_ba{^`wy~�zy^�¡UZ}ut��Z]Z}a«uxwyZ�^eux\`�{�}ux�{\x�9���9a{�µzU_9u`Z]a5ux~��9���_b�{u`�]_bX[ZS�9zyzy\=_5�9��wyZ}^©~�^r_�­�Z}\`Z]�¨~�a · w1�9z{u`Z]\õ�¨cÀ¿�Z]�4ux~�_ba ���ýÃy�ýÃ?j��9ay�µzy\`_�|9Z]^uxw{Z]~�\rX¤��uxwyZ}X��Lux~��]�9�°Z]¢?�y~�|��9��Z}ay�]Z�� sta«u`wy~�^r^`Z}�}u`~�_ba«��Z2wy~��bwy��~��bw5u��]_9X¤X[_ba���9��~�ux~�Z]^P��ay�M�y~�­�Z}\`Z}ay�]Z}^�¡UZ}ut��Z]Z}a,uxwyZ7ut��_¸�9zyz{\`_5�9��w{Z]^2��^Su`wyZ}��zUZ]\eu��9~�a�u`_uxw{ZSZ}��~��]~�ux��ux~�_ba�_����]�9�y^x���$��^`^`�{X¤z{u`~�_ba{^]�

Þ�wyZ�z{\`~�X[~�u`~�|9ZY_b¡A³=Z]�4uÆ_9�y�9a1�9���A^`~�^�~�a�uxw{Z�zU_9uxZ}a5ux~¦�����t_9�{ux�}_bX¤ZÆ�G\`�9X[Z}��_b\`´~�^/u`wyZ£�yay~�u=�t¡y�9^`Z}��\=Z]^`zU_bay^=Z¥|��9\=~¦�9¡{��Z9ª��{Z]ay_9u`Z]�Hù�c �ZÔ�j�_9\�ù 7 c-Ô¬j4ª;\`Z]�9�/ÿjàeuxw{Z|��9���{Z©uxw1�Lurù���_b�{���¯_b¡{ux�9~�av~�av�yay~�uAÔ�ª�w1�9�vø ¡¬Z}Z]a �\á'Þ�wy~�^��}_b�ya5uxZ}\=�D�9���ux�y�9��Z}a5ux~�ut��w1�9^�a1��u`�y\x�9�¥~�a5u`Z]\`z{\`Z}ux��ux~�_baM~�a,^=u`\`�y�4ux�y\`�9��Z}¢?�1��ux~�_bay^�X¤_A�yZ]��^]�· _bay^=~��yZ}\����bZ}ayZ]\`�9�±^=ux\=�y�}u`�y\x���UX[_A�yZ]��× uxw1��u��]_9a?ux�9~�a{^���^`Z4u�_9��Z]¢?�1��u`~�_ba{^

>º !Ì» º cS¼w½ º ��Ô º j�� ¿e!*d;�ÀÀÀi��ÁÂ� cÀÁ{� ì9i5j�9^¥~�a¸cÀÁ{�ýÃ?j4�Â�°Z}ujÒ!^eu��9a{���G_b\£uxwyZ�|bZ}�}ux_9\Pc�Ò����À0ÀÀ0�iÒ ä j£_��/¡1�9��´?�b\=_b�yay��|��9\=~���9¡y��Z]^}ª5��Z}u$ø��9ay�[ùסUZ�ut��_��y~�^G³=_b~�a5uÆ^`�{¡y^`Z4ux^¥_9�±_b¡y^=Z]\=|9Z]�[|9��\`~¦��¡y��Z}^]ªb��ay�F��Z}u× 7 ¡¬Zou`wyZ�^`�y¡{X¤_A�yZ]�°�}\`Z]��uxZ}�¨¡5�v\=Z]zy���9�]~�ay�7uxwyZPZ}¢A�y��ux~�_bay^��]_b\=\`Z}^`zU_bay�y~�ay�ux_µ|��9\`~��9¡y��Z}^P~�a'ø ��~�uxw�ø ! ªÆ��^�~�a ²oZ}®yay~�u`~�_ba'Á{� � ��db��Þ�wyZ¤^=ux\=�y�}u`�y\x���~�a5u`Z]\`z{\`Z}ux��ux~�_ba¯_9��ù�c �ZÔ�j�~�^��b~�|9Z]aµ¡5�

ù�c �ZÔ�j +!!ù�Ò�¡bc-Ô�j�À cÀÁ{� ì�d;jBX¢Z�Ü`_K�a�K�y\yMFky�lSagK�}�F3U^Iia¿P(UEK�WzY2FkxiFkyMI�`;F3YèL\H J�DGF1FZf'IiHGI�}�F'J�a4L)fkW�yML\J4FkagK:J�XGa�FmXGH_Y2F3a�J4D_Fa�XGvGa�LMfpS4W4UVL\J�fgDGLMHGh¸a4F3h�a�F3W�W�LMI�HÂU¬QCN?K�H_W¿PLmb3ñ�ñ0ò2~Â`s{.�*�iT�~ÂUVDGL)fgDj��F3fVP}mK�H Q-bZñ�ñ*�iTõK:J4úJ4a�LMvGX2J4FZYjJ4IAüVI:�(Q�b3ñiòGbZT|K�H5Yb£eX_K�H_YJzQ�b3ñiòQ�iT'{

Page 50: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d;ì � ÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�ÊÞ�w1��u�~�^]ªrùvc ��Ô¬j[~�^¤uxw{Z¼cD�yay~�¢?�yZ�j¤^=_b����ux~�_9a+_9�Sù¹�ya{�yZ]\�u`wyZæ\`Z]�9��~�äg��u`~�_baÒ ! Ôá~�a+uxwyZæ^=�y¡yX[_��{Z]�(× 7 _9�è×*�*½©��uxwy_b�{�bw+uxwyZ¸u`Z]\=X ç�K1IQB©~�a+uxw{ZzU_9uxZ}a?u`~¦�9� �t_b��ux�]_9X¤ZP��~�uxZ}\x��u`�y\`Z�ay_9\`X¤�9������^eu��9ay�{^©�G_b\�uxwyZ�~��yZ}a5ux~�ut�¯_��£�¤^=z¬Z���]~�®1�F~�ay�{~�|A~��y�y�9��~�a��¯zU_bzy�y����ux~�_ba/ª°�v�yay~�u�X��g������^`_¯¡UZ7uxwy_b�{�bw5u�_����9^�uxw{Z^`Z4uµ_9�2�Lu`ux\=~�¡y��uxZ]^�u`w1��u¯��w1��\x�9�4uxZ]\=~�ä}Z�u`w1��u¯~�ay�y~�|A~��y�1�9�Àª�uxwyZ�Z4��zUZ]\=~�X[Z]a5u��9��]_9ay�y~�u`~�_ba{^H�yay�{Z]\2^eux�y�{�9ªYuxwyZ[ux~�X[Z�_����1�g�bª£�9ay�M^`_æ_baA»Ü�9���£_9����w{~���w¾�9\=Z\`Z}zy\`Z}^`Z}a?u`Z]����^P�]_bX[z¬_9ayZ]a5ux^P_���u`wyZ2|bZ}�}ux_9\,Ô'~�aæ^eux\`�{�}ux�{\x�9�YX[_��{Z]��~�ay�y�Ssta�D�9�4ugªbuxw{Zr_9ay���H\`Z]¢?�y~�\`Z}X¤Z}a?u`^£_ba¸Ò!�9\`Z�cG~Qj�u`w1��u£~�uÆ\`Z]z{\`Z]^=Z]a5u���^YX��9a5�H¡1�9��´b��b\=_b�yay�+�D�9�4ux_b\=^��9^�ayZ]Z}�yZ]�Ýux_�\=Z]ay�{Z]\�u`wyZ¸\`Z}�¦��u`~�_ba{^��9X[_bay�MZ]ay�y_9�bZ]ay_9�y^|��9\`~��9¡y��Z]^F�yZ}u`Z]\=X¤~�ay~�^eux~��¯��ay�þcG~�~QjHu`w1��uFuxwyZ¨�1��ux���]_9ay^`~�^=u¤_9�P~�a{�yZ]zUZ]ay�{Z]a5u^x��X¤zy��Z]^ �y\x�g��a¸�G\`_9X �¤c�Ô¬j4�rÞ�wyZ7~��yZ]a5ux~�ut�µ_9�¥�9a¸~�ay�y~�|A~��y�1�9��zUZ]\`^=_baæ~�a¸�9aZ4��zUZ]\=~�X[Z]a5u�~�^�_9�QuxZ}a�^`��¶¤�}~�Z}a?u��G_9\�uxw{~�^�zy�{\`zU_b^`Zo¡UZ]�]�9�y^=ZP~�u�\`Z]z{\`Z]^=Z]a5ux^�uxw{Z�9a1�Lux_bX[~��]�9�{��ay�¤�9Z]ayZ4ux~���X¤�9´?~�a{�b^Æ_9�Uuxw1�Lu£~�ay�y~�|A~��y�1���Jª���w{~���w[�9\`Z�_9�Qu`Z]aF^`���¦�®1�}~�Z]a5uÆ�G_9\��yZ4uxZ]\=X¤~�ay~�a{� uxwy��uY~�a{�y~�|A~��{�1�9�Jûý^�\`Z]^=z¬_9ay^`Z�u`_©ux\=Zg��u`X¤Z}a?u`^Æ_b\Æ_�uxwyZ}\zy\=_b�b\x��X¤^�_9�$~�a5u`Z]\`Z}^=u]�

WY¢A�y��ux~�_ba*cÀÁ{� ì�d;j¤�G_b\=X¤^¯���}_bayayZ}�}ux~�_baÜ¡¬Z4ut��Z}Z]a×u`wyZ�_bz1�9¢?�yZ'WYay�b��~�^=wzywy\`�9^`Z àeuxwyZ�|9�����yZSuxw1��uoù���_b�y���«_b¡{u���~�a¸~�a¸�yay~�uõÔ�ª�wy�9�¸ø ¡UZ]Z]a á¯��ay�uxw{ZYzyw5��^=~��]�9�5zy\`_A�]Z}^`^=Z]^�uxw1��u°u`\x�9a{^=�GZ]\°��wy�9ay�bZ}^�~�aoø ~�a5ux_���w1��ay�bZ]^�~�a�ù¤��Þ�wyZ�G_b\=X���u`~�_baP_��{uxw{Z�^`�{¡yX¤_A�yZ}��× 7 Z��{z{��~��]��uxZ}^$z{\`Z]�}~�^=Z]����w{_;�MuxwyZ�w?�AzU_9uxw{Z}ux~��g�9�zywy\`�9^`Z¸à=w1�9�[ø�¡UZ]Z}a áH�}_b�y���¤¡¬Zr\`Z]�9��~�ä]Z}�/ªA�9^¥��Z]������^¥��w1�Lu�zy\=_��}Z]^=^�X2�y^eu�b~�|bZ�~�a�u`_¤X¤�9´9Zoø ! �[\=Zg�9��~�ut�9�

��~�|bZ}a�uxwy~�^�~�a5uxZ]\=zy\`Z4u���u`~�_ba¤_9��ù�c ��Ô¬j4ªA~�u¥~�^�~�ay^=u`\`�y�4ux~�|9Z u`_H�]_9a?u`\x�9^eu�uxw{ZX[Z}uxw{_��y_9��_b�b~�Z]^7_9�©�g�9�{^x�9��~�a��GZ]\`Z}ay�]Z¨~�a¾uxwyZµ�]_9�ya5uxZ]\e�D�9�}u`�1�9��|bZ}\`^`�{^�^=u`\`�y���ux�{\x�9�A�G\x�9X[Z}��_b\`´?^}�Æsâ�VÒ+~�^�ux\=Zg��u`Z]�F�9^£�©\`�9ay�y_9X*|��9\`~��9¡y��Z¥u`wyZ]a2u`wyZ�|��9���yZ�_9�uxw{Z©�]_9�ya5uxZ]\e�D�9�}u`�1�9��ùvc ��Ô¬jY¡¬Z}�]_bX[Z]^���\x��ay�y_bXÚ|��9\`~��9¡y��Zr�9^¥��Z}���Àª��yZ}ay_9u`Z]��9^©ù�c jr_b\©ù 7 � Þ�wyZ2zU_9uxZ}a?u`~¦�9� �t_b��ux�]_9X¤Z��9ay�9���A^`~�^ z{\`_A�]Z]Z}�y^P¡5�«~�X����b~�ay~�ay�uxw{Z'_9¡y^`Z}\=|bZ}�º�y~�^=ux\=~�¡y��ux~�_9aÃ��c ���ÀÀ0À0� >ä jv�9^�uxwyZ�X¤�9\`�9~�a1�9� �y~�^eux\=~�¡y�{u`~�_ba_9� �9a¼�9�y�9X¤Z}a?u`Z]�¼zy\`_9¡1�9¡y~���~�ut���G�ya{�}ux~�_baÜ� ® �{Z}®1ayZ}�Í_�|9Z]\F¡¬_�uxw¾_b¡y^=Z]\=|9Z]��9ay�þ�}_b�ya5uxZ}\=�D�9�4ux�1����|��9\`~��9¡y��Z]^]�a¤o�yZ]\=~�Z]^«�9¡U_b�{u¸�]�9�y^`�9�HZ4­�Z}�}ux^¼cQ��\`~�u`uxZ}a�¤c������ j�~�a!_9�y\v^=u`\`�y�4ux�y\`�9���9a1�9���A^`~�^xj��9\`Z�zyw{\x�9^=Z]�á�9^¯¢A�{Z]\`~�Z]^µ�9¡U_b�{u¯uxw{ZX¤�9\`�b~�a1�9���y~�^eux\`~�¡y�{u`~�_ba,_�� u`wyZ¨�}_b�ya5uxZ}\=�D�9�4ux�1����|9��\`~¦��¡y��Z�_��©~�a?u`Z]\=Z]^=u]ª���\=~�u=�uxZ}a � ® cÀùvc jâ! �{j4�!Þ�wyZ¸a{Z}��w5��zU_9u`wyZ}u`~��g���©Z]a5u`~�ux~�Z]^�ù�c j[�9\`Z¨ux\`Z]��uxZ}��9^P_b\=�y~�ay�9\=�¨\x�9a{�y_bX&|��9\=~¦�9¡{��Z]^}Û/�G_b\PZ4�{�9X[zy��Z9ª�u`wyZ}����\`ZF�9^=^`�yX[Z]��ux_v_b¡UZ}�uxw{Z¯�L��~�_bX[^H_9�rzy\=_b¡1�9¡{~���~�ut�'�g�9���]�y���y^]ªÆu`wyZv�¦�g��^2_9�r�}_bay�y~�ux~�_9ay~�ay�{ªY�9ay��uxw{Z�L��~�_9X¤^�_9�Y�}_bay�y~�ux~�_ba1�9�°~�ay�yZ]zUZ]a{�yZ]ay�}Z9��§µ_b\`Z}_;|9Z]\}ª¬uxw{Z]^`ZHw?�AzU_9uxw{Z}ux~��g�9��Z]aA�ux~�ux~�Z]^$�9\`Z���^`^`�{X¤Z}�2ux_o¡UZ��]_9ayayZ]�4uxZ}�Fu`_o_b¡y^=Z]\e|bZ]�7|��9\=~¦�9¡y��Z]^�|�~�� �}_bay^`~�^=u`Z]ay�4�

Page 51: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ�~5ÎãÒoÓeÐ�ÄPÏ�ÐyÐ�Ó4Ù�Ö d;ì�Á�]_9ay^=u`\x�9~�a?u`^�cÀê�_b¡y~�a{^Pd�÷bfbh5j¥^`�y��wæ��^ ���

ø ! ! � ùvc j�!áù�� cÀÁ{� ì � j��wy~���w¼^eu���u`Z]^7uxwy��ugªY�G_b\2Z4|bZ}\=�ÜÔ�ªY~���uxw{Z¯�9�}u`�1�9��|��9���yZ�_9�rø ux�y\=ay^F_b�{u2u`_¡UZ ª±uxwyZ}a¨u`wyZ�|��9���yZPuxw1��u©ù"��_b�{���¨ux�9´9ZS_ba¨~���ø ��Z]\`Z ~�^�Z]¢?�1����u`_¤uxw{Z�9�4ux�1�9��|9�����yZ�_��où[�,Þ�w?�y^]ª¥��wyZ]\=Zg�9^7uxw{Zµ^eux\=�y�}u`�y\x�9���9zyzy\=_5�9��w¾|�~�Z}��^7uxw{Z~�a5u`Z]\=|9Z]a5ux~�_ba¤êsë�c jF�9^¤�9aÍ_9z¬Z}\x��u`~�_ba¾uxw1�Lu¤��wy�9ay�bZ}^¤uxw{Z«X[_��yZ}�ScÀ�9a{�Íuxw{Z�y~�^=ux\=~�¡y��ux~�_9a±jS¡y��u7´9Z}Z]zy^¤�9���£|��9\`~��9¡y��Z]^Hu`wyZv^x�9X[Z9ªÆu`wyZvzU_9uxZ}a?u`~¦�9� �t_b��ux�]_9X¤Z�9zyz{\`_5�9��w�|A~�Z4��^Su`wyZ7|9��\`~¦��¡y��ZHù �{ay�yZ]\,êsë{c j ux_µ¡¬ZF�¯�y~�­�Z}\`Z]a5u�|��9\`~��9¡y��Z9ªù�c j4ªb��_�_9^`Z]���P�]_9ayayZ]�4uxZ}�2ux_PùÝuxwy\=_b�y�bwH\`Z]����ux~�_bay^�^`�{��wF�9^�cDÁ{� ì � j4��sta · w1�9zA�uxZ}\j�7��Z�^=wy_���ª¬�{^`~�a{�Fu`wyZS^=ux\=�y�}u`�y\x����~�a5uxZ]\=zy\`Z4u���u`~�_ba¯_9��ùvc ��Ô¬j4ª{u`w1��ur~�u�~�^~�a{�yZ]Z}�ã��Z}�b~�u`~�X¤��uxZ¤ux_�ux\`Z]��u¤�]_b�ya5u`Z]\=�D���}ux�y�9��^¤�9^[\x�9ay�{_bX |��9\`~��9¡y��Z}^F~�aÍ�9���\`Z}^`zUZ]�4ux^S�9ay�/ª°X[_b\=Z]_�|bZ}\]ª/uxwy��uP�]_9ay^`~�^=uxZ}ay�}�¸�]_bay^eux\`�9~�a5ux^©��~�´�Z¯cÀÁ�� ì � jr�G_b����_���9^�uxw{Z]_b\=Z]X[^��G\=_bX�u`wyZ�^=u`\`�y�4ux�y\`�9�°~�a5uxZ}\`zy\=Z}ux��ux~�_ba/�

Þ$_��]_bX[X2�yay~��g��u`Z�^=�y¡y^eu��9a5ux~�|bZ��]�9�y^`�9�Í´?ay_�����Z]�y�bZ�ª!u`wyZ�zU_9u`Z]a5ux~��9���_b�{u`�]_bX[Z��9a1�����A^=uµX2�y^euµZ4��zy\`Z}^`^¨�g�9�y^`�9���9^=^`�yX[z{ux~�_bay^µ�9^¯�]_9ay^=u`\x�9~�a?u`^¯_ba� ® ª9�y^=�1�9������~�a�uxw{Z��G_b\=Xþ_9�1�]_9ay�y~�u`~�_bay�9�5~�ay�{Z]zUZ]ay�yZ}ay�]Z��9^`^=Z]\=u`~�_ba{^�~�a5|9_b��|A~�a{��]_9�ya5uxZ]\e�D�9�}u`�1�9�5|��9\`~��9¡y��Z]^]��è1_b\�Z4�{�9X[zy��Z9ª;ux_r�]_bX[X2�yay~��g��u`ZÆuxwyZ¥�yay�{Z]\`^eu��9a{���~�a{��uxwy��ue»Ý~�a¤�H\x�9a{�y_bX[~�ä]Z}���}��~�a{~��g���yux\`~��9�y��~�u`w�~�X[zUZ]\=�GZ}�}u��]_bX[zy��~¦�9ay�}Z�cD^`Z}Zè�~��b�y\`Z[Á{�M�ycG¡±j`jJ»Ýuxw{Z[���g��^=�y¡A³=Z}�}ux^H\`Zg���}u¯cDù2joux_µux\`Z]��uxX[Z]a5u`^�cQø'j�~�^�^=u��;�ux~�^=u`~��g��������~�ay�yZ}z¬Z}ay�yZ}a?u�_��YuxwyZHu`\`Z]��uxX[Z]a5uP��^`^`~��bayX[Z]a5u[cÀúoj4ªUuxwyZ2zU_9u`Z]a5ux~��9���_b�{u`�]_bX[Z¯�9a1�����A^=u7��_b�y���¾��\=~�u`Z¨ù�c j1:G:�ú��Ü�°~�´9Z4��~�^=Z9ª�u`_��]_ba5|bZ4��uxwyZ¯�yaA��yZ}\`^=ux�9ay�y~�ay�Hu`w1��u�u`wyZP�9^=^`~��bayX[Z]a5u�~�^�\x��ay�y_bX[~�ä}Z]�v�9a{��wyZ]a{�]ZP~�a{�yZ]zUZ]ay�{Z]a5u_9�Swy_���^`�y¡?³=Z]�}u`^¨�}_bX[zy���¾��~�u`w+uxwyZ¸�9^`^=~��bayX[Z]a5u]ªru`wyZ¸zU_9uxZ}a?u`~¦�9� �t_b��ux�]_9X¤Z�9a1�����A^=u���_b�y���¨�y^=ZPuxwyZ�~�a{�yZ]zUZ]ay�{Z]ay�}ZH�]_9ay^=u`\x�9~�a?u ú¥:G:Pø¼c4/5j��

½ã�}_b����Z]�}u`~�_baS_9�±�}_bay^=u`\x�9~�a5ux^�_��1uxwy~�^�ut�AzUZ�X¤~��bw5u�^=_bX[Z}ux~�X¤Z}^$¡UZ�^=�{¶[�]~�Z]a5uux_�z¬Z}\`X[~�u[�'�yay~�¢?�yZµ^`_9���{u`~�_ba�ux_'uxwyZµ¢?�yZ]\e�ã_��o~�a5u`Z]\`Z}^=u]Û ~�aÍ_9u`wyZ]\[�g��^`Z]^}ª_bay���¯¡¬_9�yay�y^ _ba¨u`wyZH^=_b���{u`~�_baµ�g��a«¡UZH_b¡{ux�9~�a{Z]�/��è1_b\rZ4�{�9X[zy��Z�ª±~��Æ_9ayZH�]�9azy���9�y^`~�¡y���Í�9^=^`�yX[Z«u`w1��u¯�,^=Z}u¨ú�_9�S�]_�|��9\=~¦��u`Z]^�^`��ux~�^=®1Z}^�uxwyZæ�}_bay�y~�ux~�_9a1�9�~�a{�yZ]zUZ]ay�{Z]ay�}Z

ùvc j1:G:Pø¤� ú cÀÁ{� ì9Á5jcÀ��a¨��^`^`�{X¤z{u`~�_bavu`w1��u����9^�uxZ]\=X¤Z}�¢à=�]_ba{�y~�u`~�_ba1���¬~��bay_b\`�9¡y~���~�ut��áH¡5�'cDêr_9^`Z]aA�¡1�9�{X��9a{�7êr�{¡y~�a¤d�÷bfbÁ5j�ª�u`wyZ]a2u`wyZ��]�9�y^x���{Z}­¬Z]�4u�� ® cÀù�c j�!��{j��]�9a2\`Z]�9�y~����B4BkôÂL\v_v_K�agY·K�H_Y;�eK�a�`5F3a1Q-bZñ0ý9�_~s`s{^bZò$�0TÂF'�2`Ga�F3W�W�FZY(J�DGL)Wzf'IiH_W�J4agK�LMHiJ�KiWQ¦¨§ þMQ<¦zÿ��© T�ª © �û{

Page 52: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d;ì�Ã1ÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê¡UZSZ4|��9���1�LuxZ]��ªy�y^`~�ay�¨cÀÁ{� ì � j�ª{ux_F�A~�Z}��� ���

� ® cDù�c j�! �yjç! #�$Õ� ® cÀù�c j�!����</5jk��c�/5j! # $ � ® cÀù�c j�!���� �:/5jk��c�/5j! #�$ � ® cÀù ! ��� ��/5j3�¤c4/5j! #�$ �¤c���� �:/5jk��c�/5j�À cÀÁ{� ì�Ã?j

Þ�wyZr�¦�9^eu¥Z4��zy\`Z}^`^=~�_ba��}_ba5u��9~�ay^£ay_H�}_b�ya5uxZ}\=�D�9�4ux�1���U¢?�1��a?u`~�u`~�Z]^©cGu`w?�y^�zUZ]\=X¤~�u=�ux~�ay���{^ ux_��y\`_bz¨uxwyZH�9^=u`Z]\`~�^`´¯�G\`_bX � ® jr�9ay�¸�}_b~�ay�}~��yZ}^©zy\=Z]�}~�^`Z}���¯��~�u`w«uxw{Z�9�L³=�{^=uxX[Z]a5u[�G_b\`XH�y�¦�¸_9�7cÀÁ{��d�÷5j�ª¥��w{~���wã_9¡{u��9~�ay^7�G\`_bX uxwyZµ¡1�9��´b�t�{_�_b\F�}\`~��uxZ}\`~�_ba/�,³ _���Z4|bZ}\]ª�uxwyZF�9^=^`�yX[z{u`~�_baæ_9�¥�}_bay�y~�ux~�_9a1�9��~��bay_9\x�9¡y~���~�ut�'cDZ}¢A�y��ux~�_bacÀÁ�� ì9Ábj`jâ»+uxwyZS´9Z4�«u`_�uxwyZH�yZ]\=~�|���ux~�_baµ_9� cÀÁ{� ì�Ã?jJ»Ü~�^ ay_�uo^=u`\x�9~��bw5u`�G_b\e����\`�¨u`_�]_9X¤zy\=Z]wyZ}ay�F_b\Y�9^=�]Z}\=u���~�a/�´��9\`�9zywy\`�9^`Z}�¤~�a2Z4��zUZ]\`~�X¤Z}a5u��9�yX[Z}ux�9zywy_9\`^]ª�uxwy~�^�9^=^`�yX[z{ux~�_baH\=Zg�9�{^]ÿ�Þ�wyZ������H�9a�~�ay�y~�|A~��y�1���b��~�uxwH��u`u`\`~�¡y�{uxZ}^Yú���_b�{���2\=Zg�9�4uux_[ux\=Zg��u`X¤Z}a?u øÆ! ~�^ ~�ay�yZ}z¬Z}ay�yZ]a5uP_��ÆuxwyZSux\=Zg��u`X¤Z}a?u©�9�4ux�1�9�����¯\=Z]�]Z}~�|9Z]�¡5��uxwy��ur~�ay�y~�|A~��y�1���J�

¿�Z]�4ux~�_ba[Á{� h�� � Z���zy�¦�9~�ay^���w5�2u`wy~�^Y��zyzy\`_b�9��wFX��g�H�9zyzUZg�9�{u`_P^`_bX[Z�^eu���u`~�^e�ux~��]~��9ay^]ªrZ}|9Z]a×u`wy_b�y�9wÜu`wyZ�zy\`_A�]Z}^`^µ_9��Z]��~��]~�ux~�a{�µ³=�y�y�bX[Z]a5u`^¨��¡¬_b��u¯�]_b�yaA�uxZ}\=�D�9�4ux�1�����yZ]zUZ]a{�yZ]ay�}~�Z}^¤w1��^7¡¬Z}Z]a¾Z4�Aux\=Z]X[Z]���,�y~�¶¤�}�y��u7�9ay�¾Z]\=\`_b\O�tzy\=_bayZ9Û~�a{^=uxZ]�9�¼_9�o�}_bay^=u`\`�y�4ux~�ay�MayZ4��|b_A�g��¡y�y�¦��\=�¾�9a{�ÍayZ4� ��_b�9~����G_b\F�]�9�y^x���rZ��?�zy\=Z]^`^=~�_ba{^]ªy�����±X��LuxwyZ}X���u`~��]�9�y_bzUZ]\`��ux~�_bay^¥~�aFuxwyZ z¬_9u`Z]a5ux~��9���â_b�{u`�]_bX[Z��G\x�9X[Z4���_b\`´ã�9\`Zæ�}_bay�y�y�4uxZ}�!��~�uxwy~�a+uxwyZæ^`���GZæ�]_ba�®1ayZ]^¯_9�Szy\=_b¡1�9¡{~���~�ut�¾�g�9���]�y���y^]�Þ�wyZ[�y\x�g��¡1����´���~�Z]^P~�a'u`wyZ[\`Z]¢?�y~�\`Z}X¤Z}a?uS_9���y^`~�ay�µ~�ay�yZ}z¬Z}ay�yZ}ay�]~�Z]^2��X¤_ba{��]_9�ya5uxZ]\e�D�9�}u`�1�9�5|��9\`~��9¡y��Z]^°ux_rZ4��zy\=Z]^`^$zy�¦��~�a��g�9�{^x�9�?´?ay_�����Z}�y�bZ9�$åÜwyZ]a��]_b�yaA�uxZ}\=�D�9�4ux�1���/|9��\`~¦��¡y��Z}^��9\`Z�ay_�u�|�~�Z}��Z]�«�9^�¡?�Azy\=_��{�y�}u`^ _9���7�yZ]Z}z¬Z}\]ª±zy\=_��}Z]^`^O�¡1�9^=Z]��X[_A�yZ]�ÀªA~�u�~�^�wy�9\`�[ux_2��^`�]Z}\=ux�9~�a���w{Z}uxw{Z]\�>9CQCb\=Z]��Z}|��9a5u��}_b�ya5uxZ}\=�D�9�4ux�1���~�a{�yZ]zUZ]ay�{Z]ay�}Z¯³=�y�y�9X¤Z}a?u`^¯w1�g|bZM¡UZ]Z}a×��\=ux~��]�y����uxZ}�/ª �� ��w{Z}uxw{Z]\¯uxwyZ�³=�y�y���X[Z]a5ux^ �9\=u`~��}�y�¦��u`Z]�µ�9\=Z�\=Z]�y�ya{�1�9a5ugªU_b\���wyZ}u`wyZ]\ruxwy_9^`Z�³=�y�y�bX[Z]a5ux^ �9\=Z�^=Z]���¦��]_9ay^`~�^=uxZ}a5ug��Þ�wyZ¸Z}��~��}~�ux��ux~�_baÍ_���^`�y��wÜ�]_9�ya5uxZ]\e�D�9�}u`�1�9�¥³=�{�y�bX[Z]a5ux^��]�9aÝ¡UZ^=�A^euxZ]X¤��u`~�ä]Z}��¡5�æ�{^`~�a{�¯u`wyZ7�G_b����_���~�ay��u`\x�9a{^`�¦�Lux~�_9a«�G\`_bX&�b\x��zywy^¤cD^=Z]Z�¿�Z]���ux~�_baâ����db�ÂÃ7�G_b\����y�y~�u`~�_bay�9�¬\=Z]����ux~�_bay^`w{~�zy^xj4�B-ìZôÂL\v_v_K�agY¶K�H_Y«�.K�a4`;FkaÜQ�b3ñ0ý$�G~ `s{�bZò0ý�TâX_W4F3Y J4DGFóS4L\hiHGI�agK�v_L\yMLdJ-� KiW4W4XG}�`2J�L\IiHbUø1Q r T�¬�¬e��J4IAY2F3a4LMx�FeJ4D_F�F3�0X_K�yML\J-�1NmQ8øjQ r T ù RGT ù NmQSRsn r T�{B�M3�âJ-�0`_LMf3K�y�I:x�F3a�W4L\hiD0J�L\HjJ4D_FÂFk�GK�}�`GyMFeI�lµY�L\hiXGa4F��G{ ý2Q8v5T�D_K�W|v;FkFkHjJ4I�UVa�LdJ�F*­®¬�¬eøjQ r TK�H5Yb­®¬�¬e� Q<¯TwLMH_W�J4F3KiYõI�lC­®¬W¬�°ZøAQ r T|±4�ÝQ<¯T�²0~GK�WVY2L)f�JgK:J�F3Yõv�?Q8�_{ òQ�iT�{

Page 53: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ�~5ÎãÒoÓeÐ�ÄPÏ�ÐyÐ�Ó4Ù�Ö d;ì9ì��\`�9zywy^�Z}ay�]_A�yZ[^`�y¡y^eu��9a5u`~�|bZ[~�a��G_b\`X¤��ux~�_ba«~�a�¡¬_9u`w'u`wyZFZ]¢?�1��u`~�_ba{^���ay�

uxw{Zrz{\`_b¡1��¡y~���~�ut�S�G�yay�}u`~�_ba �¤c�Ô¬j4Ûbu`wyZ��G_b\`X[Z]\Y~�^¥Z}ay�]_A�yZ}���9^£X[~�^`^=~�ay����\`\`_���^}ªuxw{Z7�¦��u=uxZ]\o�9^PX¤~�^`^=~�ay���y�9^`wyZ}�'�9\`�}^]�7W£�9��w�z1�9\=Z]a5u=�â��wy~����¸�D�9X[~����Mc��^f º �`ø º j~�a¸���g�9�y^`�9�Æ�y~��9�b\`�9XÖ±��]_9\`\`Z}^`zU_bay�y^Pu`_¯�9aæZ]¢?�1��u`~�_baæ~�a«uxw{Z7X¤_A�yZ]�Â× _9�cÀÁ�� ì9ibj4�¬³ Z}ay�]Z�ªYX¤~�^`^`~�ay�¨�9\`\=_;��^�Z]ay�}_��{Z�Z4���]���y^`~�_ba,��^`^`�{X¤z{u`~�_ba{^]ª�u`w1��u�~�^]ª�]���9~�X[^ouxwy��u��9�y�y~�ay�¨Z����]���{�yZ]�M|��9\=~¦�9¡{��Z]^Pu`_«�9a�Z]¢?�1��u`~�_ba���~����Æay_9u���w1�9a{�bZuxw{Zµ_9�{ux�}_bX¤Z�_9� u`wyZ¯w5��zU_9u`wyZ}u`~��g����Z4��zUZ]\`~�X¤Z}a5u¤�{Z]^`�}\`~�¡¬Z}�Í¡5�,u`w1��uFZ]¢?�1�;�ux~�_ba/��§¨~�^`^`~�ay�¸�1��^`wyZ}�ã�9\=�]^7Z}ay�]_A�yZv~�a{�yZ]zUZ]ay�{Z]ay�}~�Z]^F��X¤_ba{�¸�y~�^eux�y\=¡1�9ay�}ZuxZ}\`X[^�~�a�ut��_¨_b\�X¤_b\=ZFZ]¢?�1��u`~�_bay^}�vè1_b\PZ4�{�9X[zy��Z9ª�uxwyZ¤�9¡y^=Z]ay�}Z�_9���1�9^`w{Z]��9\=�]^7¡UZ}ut��Z]Z]aã�¸ay_A�yZ¯ù �9ay�¼�¸^`Z4u[_9�ray_��{Z]^¸Þ�ú����ÀÀÀi��ú·¶_ߨ~�X[zy��~�Z]^�uxw1�Luuxw{Z¨�}_b\`\=Z]^`zU_bay�{~�ay�'¡1�9��´?�b\=_b�yay�ã|��9\=~¦�9¡{��Z]^}ªAÒeÐ��9a{� Þ�Ò®) & �ÀÀÀi�iÒ®)�³2ß?ª��9\=Z~�a{�yZ]zUZ]ay�{Z]a5u©~�a¬��c-Ô�j��

Þ�wyZ]^=Z,�9^=^`�yX[z{u`~�_bay^¯�]�9aסUZæux\x��ay^`����uxZ}�á~�a?u`_�uxwyZ�zU_9uxZ}a?u`~¦�9� �t_b��ux�]_9X¤Zay_9ux��ux~�_ba¯�y^`~�ay�7ut��_¤^`~�X¤zy��ZP\`�y��Z]^�c4´�Zg�9\=�Yd�÷b÷5ì��{ªyz/�^�9i9Ã5jezUZg�9\=�JÿÂ÷5ì�Û�u`wyZP®1\`^eu~�a5u`Z]\`z{\`Z}u`^vuxwyZæX[~�^=^`~�a{���9\`\=_���^µ~�a+uxwyZæ�9\x�9zyw�ª uxw{Zæ^`Z]�}_bay�/ª uxwyZ�X[~�^`^=~�ay��1�9^=wyZ]�«�9\`�}^]�db�b´�<�@4CEç5:4IJ6LK«ßx0�:4BÀß�IJ@4BDIJ6LK{:�ü¥è1_b\�Z}|bZ}\=��|��9\=~¦�9¡{��Z ù w1�g|�~�ay�7z1�9\=Z]a5ux^A�gf¥µ�9a{�µ�G_b\�Z4|bZ}\=�¯^`Z4ur_9�$|��9\=~¦�9¡{��Z]^ �¾�y~�^Q³=_b~�a5u�_����gf¥µ�ªy��ZSw1�g|bZ

ùvcS¼>½�µ�j�!áù�cS¼w½�µÂ�9�Lj�À cÀÁ{� ìbìbj

� �OÛ�KUTA0>=104KUTA04K�@�0[ß�0�:4BDß�IJ@4BÀIJ6LK{:�ü[sâ�oú����À0ÀÀ0��ú·¶�~�^2�9a5�M^`Z}u2_9�ray_��{Z]^2ay_�u�}_bayayZ}�}uxZ}�¨u`_¤ù |A~¦�7�1��^`wyZ}�¸�9\=�]^]ªy��Z�w1�g|bZ � �

ù�c%¼w½�µ/j1:G:ÝÞ�ú��}c%¼w½ B & j��À0ÀÀ0��ú·¶9cS¼w½ B ³ j:ßsÀ cÀÁ{� ì9h5j

Þ�wyZº~�ay�yZ}z¬Z}ay�yZ]a{�]Zþ\`Z}^=u`\`~��4ux~�_ba�ux\`�9ay^=�¦��u`Z]^Íu`wyZº~�ay�yZ]zUZ]a{�yZ]ay�}Zþ¡¬Z4ut��Z}Z]aÒeÐ!�9ay� Þ�Ò®) & �ÀÀÀi�iÒ®) ³ ߯~�a5ux_¨~�ay�{Z]zUZ]ay�yZ}ay�]Z�¡¬Z4ut��Z}Z]a�uxwyZ¤�]_b\=\`Z}^`zU_bay�y~�ay�zU_9uxZ}a?u`~¦�9� �t_b��ux�]_9X¤Z[|9��\`~¦��¡y��Z}^]�,Þ�wy~�^7�G_b����_;��^2�G\=_bX u`wyZ¯_b¡y^=Z]\=|���u`~�_ba�u`w1��ugª_bay�}Z���Z�^`Z}u�uxwyZ}~�\�zy�9\`Z}a?u`^]ª5uxw{Z�|��9\`~��9¡y��Z}^$~�aÑÞLù���ú��:�ÀÀÀ���ú·¶5ß�^=ux�9ay�7~�aH�G�yay���ux~�_ba1�9�¬\`Z}�¦��u`~�_bay^=wy~�z{^�ux_7uxwyZ Òþu`Z]\=X¤^�~�a¯u`wyZ]~�\��]_b\=\`Z}^`zU_bay�y~�ay�¤Z}¢?�1��ux~�_bay^}�B � CEDGFÈa4FZW-J�a4L)f�J�L\IiHÃLMWâL\H l8K�f�J W�J4a�I�HGhiFkaZ~ �-IiL\H0J4yM�ÃK�`G`_y\�LMHGhÜJ�I¢K�yMyèL\H_W�J�K�H0J4L)K:J4LMI�H5WI�l J�DGF¢N*¦"x:K�a�LMK�vGy\FZWk{ Y_I�aÈF'�GK�}�`GyMF�~ �;¬�¬eø1QN¶L·�¸�TÄW4DGI�XGy)Y v;F©LMH0J4Fka�`Ga�F'J�F3Y KiW�p¬W¬Q°ZøAQN¶L· � ¸ T2±4øjQ¹¶L· � �¸ T|±�ø1QN¶L· � � �¸ T|±1º#º1ºN²0~:UVDGFka�F»¶�· � ¸ ±<¶�· � �¸ ±G¶L· � � �¸ ±#º1º#º�K�a�FwJ�DGF^x:K�y\XGFZWsJ�D_K:J�J4D_FW4F'J.N*¦¼¸·}�K��AJ�K$PiF�I�Hs{

Page 54: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d;ì9hyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�ʽ©^Y��aHZ4�{�9X[zy��Z9ª�u`wyZ�X[_A�yZ]�A^`wy_���a7~�a2è�~��b�y\`Z¥Á{� ì �y~�^`zy���g��^�uxw{Z��G_b����_���~�ay�

z1�9\=Z]a5ur^`Z4ux^]ÿ�^f ? ! Þ¾½�ßs�A�gf B ! Þ�ø ßs�A�gf µ ! Þ�ú?ßsÀ cÀÁ{� ì��bj

· _bay^=Z]¢?�yZ}a?u`���9ªUu`wyZ�Z4���]���y^`~�_ba¯\`Z]^eux\=~��}u`~�_ba{^�ux\x��ay^`����uxZP~�a?u`_yÿú2c j"! ú2c���� j�� cÀÁ{� ì9f5jø¼c-�yj ! ø¼c4/��Z�yj�!!ø¾c4/5j�!Üø�� cÀÁ{� ì9÷5jùvc4/bj"! ùvc4/�� j4Û cÀÁ{�Âhbi5j

uxw{ZH�9¡{^`Z]a{�]Z2_9�Æ�¤�1��^`wyZ}�æ�9\`��¡UZ}ut��Z]Z}a'ú*�9ay�ÓÞLùz�=ø ßHu`\x�9ay^=�¦��u`Z]^�~�a5u`_¤uxw{Z~�a{�yZ]zUZ]ay�{Z]ay�}Z�\=Z]^=u`\`~��}ux~�_ba

úHc j�:<:ÝÞLù�c4/5j��`ø ßsÀ cÀÁ{�Âh{d;j�P~�|bZ}a �º^`�{¶[�]~�Z]a5uÍaA�{X2¡¬Z}\�_9�µ^`�y��w�\`Z}^=ux\=~��4ux~�_9ay^,_ba¹� ® ª�uxwyZÜ��a1�9���A^eu��u=uxZ]X[z{u`^�ux_×�]_bX[zy�{u`Z+�g�9�{^x�9�¤Z}­¬Z]�}u`^Ü� ® cÀùvc j ! �{jæ�y^`~�ay�þ^=ux�9ay�1�9\=�zy\=_b¡1�9¡y~���~�ut�¸�g�����]�{���y^�ux_b�9Z}uxw{Z]\���~�uxw'uxwyZ¤��_b�9~��g���Æ�]_ba{^=ux\`�9~�a5u`^�cGZ9�ý�y��cDÁ{� ì � j`juxwy��u¸�}_b�yzy��Z,�]_b�{a?u`Z]\e�D�9�}u`�1�9�H|��9\=~¦�9¡y��Z]^¯��~�u`w×uxwyZ}~�\¨X¤Z]�9^`�y\`�9¡y��ZM�]_9�ya5uxZ]\O�z1�9\eux^}��Þ�wyZ]^=Z��]_ba{^=ux\`�9~�a5u`^¸�g�9aס¬Z,�{^`Z]����^¸�L��~�_9X¤^}ªS_9\«\`�{��Z]^¨_9�F~�a��GZ]\e�Z]a{�]Z9ª°~�aæ��u=uxZ]X[z{u`~�ay��u`_�ux\`�9ay^=�G_9\`X��g�9�y^`�9��Z4­�Z}�}u�Z��{z{\`Z]^=^`~�_bay^�_9�£uxwyZ��G_b\`X� ® cÀù�c jÄ! �{j�~�a5ux_�Z���zy\`Z}^`^`~�_bay^µ~�a5|b_9��|A~�ay��_9ay���ãX¤Z]�9^`�y\`�9¡y��Z«|��9\`~��9¡y��Z}^]�åÜwyZ}a�^=�y��wM��u`\x�9a{^=�G_b\=X���u`~�_ba«~�^©�G_b�{ay�/ª/uxw{Z7�]_b\=\`Z]^=z¬_9ay�y~�a{�µ�]�9�y^`�9�ÆZ}­¬Z]�4u~�^�~��yZ]a5u`~�®±�9¡{��Z9ª�^`~�ay�]Z?� ® u`wyZ]aµ\`Z}�y�y�]Z}^ru`_Ñ�F�

Þ�wyZ«¢?�yZ}^=u`~�_ba+ay��ux�y\`�9�������9\=~�^=Z]^[_9�o��wyZ4uxwyZ}\¤uxw{Z¨�}_bay^=u`\x�9~�a5ux^��{^`Z]�ã¡?�zU_9uxZ}a?u`~¦�9� �t_b��ux�]_9X¤Z¥�9a1�����A^=u`^Y�9\`ZS@�6LR/=UC�04BO0â»+uxw1��u$~�^}ª9��wyZ4uxwyZ}\ÆuxwyZ4�2�9\=Z�^`���¦�®1�}~�Z]a5u��G_9\��yZ]\=~�|A~�a{�2Z}|9Z]\=��|9����~��¤^=ux��uxZ}X¤Z}a?ur��¡¬_b��u��g���y^x�9��zy\`_A�]Z}^`^`Z}^]ª±~�a5u`Z]\e�|bZ}a5ux~�_9ay^]ª��9ay�«�}_b�ya5uxZ}\=�D�9�4ux�1����^]��Þ�_���ay^=��Z]\ruxwy~�^ ¢?�yZ}^=ux~�_ba/ª±uxwyZ�|��9��~��y~�ut��_9��]_9�ya5uxZ]\e�D�9�}u`�1�9�±^=ux��uxZ}X¤Z}a5ux^�a{Z]Z]�¤¡¬Zr�yZ4®1ayZ]��\`Z]����ux~�|bZ�ux_SX[_b\`Z�¡1�9^=~���X¤��uxw��Z]X¤��u`~��g����_b¡A³=Z}�}u`^]ª�^=�y��w,��^SzU_b^=^`~�¡{��Z2��_b\`���y^¤cJ¿�Z}�}ux~�_ba¼db�ÂÃ{�ÂÃ?j©_9\P^=ux\=�y�}u`�y\x���Z]¢?�1�Lux~�_9ay^ScDZ}¢A�y��ux~�_ba�cÀÁ{� ì�d;j=j4�Æsta¯uxwyZP^eu��9a{�1�9\`�µzU_9uxZ}a?u`~¦�9� �t_b��ux�]_9X¤Z��G\x�9X[Z4���_b\`´±ª©w{_;��Z}|9Z]\}ªouxwyZ�¢?�yZ]^eux~�_baÜ_9�S�]_bX[zy��Z4uxZ}ayZ]^=^µ\=Z]X¤�9~�a{^�_bzUZ]a/ª ¡¬Z}�g�9�y^=Zù�c ��Ô¬j[~�^�u��9´�Z]aÜ�9^��Mzy\`~�X¤~�ux~�|bZµay_9ux~�_ba+��ay�Ý¡¬Z}�g�9�{^`Zæ�]_9ay^`~�^=uxZ}ay�}�ã�]_baA�^=u`\x�9~�a5ux^P^`�{��w���^[cÀÁ{� ì � j©����uxw{_b�y�bw«u`wyZ}�æ�9zyzUZg�9\oz{�¦�9�y^=~�¡y��Z��G_b\ uxwyZFWYay�b��~�^=wZ4��zy\=Z]^=^`~�_9a à=wy�9�7ø&¡UZ]Z}a á;»!�9\=Z�ay_9uY�yZ}\`~�|bZ]�F�G\`_9X �P�yZ]Z}z¬Z}\£X���u`wyZ]X¤��u`~���g���y_b¡A³=Z}�}ug�YÞ�w{~�^Y¢?�yZ]^eux~�_baF_9�U�]_bX[zy��Z4uxZ}ayZ]^=^¥~�^Y^`Z4u`ux��Z]�F~�a · wy�9z{uxZ}\���ªb��wyZ]\=Z

Page 55: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ�~5ÎãÒoÓeÐ�ÄPÏ�ÐyÐ�Ó4Ù�Ö d;ì;���ayZ}�]Z]^=^x�9\e���9ay��^=�{¶[�]~�Z}a5u�^`Z}u�_����;�{~�_bX[^©~�^©�yZ]\=~�|bZ}�æ�G\`_bX uxwyZH^=ux\=�y�}u`�y\x���^`Z}X���a?u`~��}^r�9~�|bZ}avux_¤ù�c �ZÔ�j�¡5��cDÁ{� ìAd;j4�sta¸�9^=^`Z}^`^`~�ay��u`wyZHwy~�^=u`_b\`~��g�9�/�{Z}|bZ}��_bz{X¤Z}a?u©_9�Æ^eux\=�y�}u`�y\x�9��Z]¢?�1�Lux~�_9ay^©��ay�zU_9uxZ}a?u`~¦�9� �t_b��ux�]_9X¤Z[X¤_A�yZ}��^]ª�_bayZ��]�9ayay_9uF_�|9Z]\`Z}X¤z{w1�9^`~�ä]Z�u`wyZ�~�X[zU_b\=ux�9ay�]Z_9� uxwyZ¯�]_9ay�]Z}z{ux�1��� �}�¦�9\=~�ut�Mu`w1��u[^=u`\`�y�4ux�y\`�9��Z]¢?�1��u`~�_ba{^[_�­�Z}\F|A~�^e�À¿�;�â|A~�^7uxw{ZzU_9uxZ}a?u`~¦�9� �t_b��ux�]_9X¤ZPX[_��{Z]�J��Þ�w{Z2\`Z]�9�yZ]\ X��g�µ�9z{zy\`Z}�]~¦�LuxZ�u`wy~�^r~�X[zU_b\=ux�9ay�]Z¡5�[��u`u`Z]X[z{ux~�ay�oux_�³=�y�{�bZr��w{Z}uxw{Z]\£uxwyZ��]_9ay�y~�u`~�_ba7_��£cÀÁ�� h{d�j�wy_b���{^Y~�a[�S�9~�|bZ}a�D�9X[~���~¦�9\�^=~�ux�y��ux~�_ba/�oÞ�wy~�^o�]_9ay�y~�u`~�_ba«\`Zg���y^]ÿÑàeuxw{Z2|9�����yZSuxw1��u�ú*��_b�{���æ_b¡��u���~�a�wy�9��ø�¡UZ]Z}a ~�^°³=_b~�a5ux���7~�a{�yZ]zUZ]ay�{Z]a5u�_9�°¡U_9u`w�ø �9ay�¤uxwyZr|9�����yZ�uxw1�Luù ��_b�y���«_b¡�u��9~�a«w1�9�æú¡UZ]Z]aÄ/��\á�cGsta«uxw{ZH^=u`\`�y�4ux�y\`�9��\`Z]z{\`Z]^=Z]a5u��Lux~�_9a/ª�uxw{Z^`Z}a5uxZ]a{�]Z[\`Zg���y^]ÿ¸àxú�^`w1��\`Z]^�ay_v�]�9�y^=Z2��~�uxwæZ}~�u`wyZ]\©ø _9\Pù¤ª°Z��{�}Z]z{uP�G_9\©ø~�u`^`Z}���Oª1�9^r^`w{_;��a¨~�a¨è�~��9�y\`Z�Á{� ì��dá?j[Þ�w{ZSu`wy_b�y�bw5ur_9��w1�g|�~�ay�Fux_[Z4��zy\`Z}^`^}ª��yZ���GZ]a{�/ª°�9ay��X¤�9a1�9�9Z��G_b\=X¤~��1�9¡y��ZS�}_b�ya5uxZ}\=�D�9�4ux�1���Æ\`Z}�¦��u`~�_bay^=wy~�z{^©_9�Yu`wy~�^©ut�AzUZX¤���¯Z���zy�¦�9~�aµ��w5�¯uxw{Z�Z}a?u`Z]\=zy\`~�^`ZH_��Æ�g���y^x�9��~�a��GZ]\`Z}ay�]ZS~�^��}�y\`\=Z]a5ux���µ|A~�Z}��Z]���~�u`wÜ^`�y��w×����Z��9ay�!�yZ]^=z1�9~�\¯��X¤_ba{�¼\`�9ay´b�O��ay���â®y��Z¸Z}zy~��yZ}X¤~�_b��_9�b~�^eux^���ay�^=ux��ux~�^=u`~��]~��9ay^t»!�9ay�[��w5�vZ]�}_bay_bX[~�^eux^���ay�v^`_A�]~��9�¬^=�]~�Z]a5ux~�^=ux^��}_ba5ux~�aA�{Z©ux_7�y^=Z^=u`\`�y�4ux�y\`�9�?Z]¢?�1��u`~�_ba{^$~�ay^=u`Zg�9�S_9�{u`wyZ¥z¬_9u`Z]a5ux~��9���â_b�{u`�]_bX[ZY�9��uxZ]\=a1��u`~�|bZ}^��9�{|b_L��g�LuxZ]�H~�a,³ _b���¦�9ay�¤cOd�÷bfbf5j�ª�½ ay�b\=~�^=u$Z}u��9�À�Aced�÷b÷9h5j4ª��9ay�2¿�_b¡UZ]��ced�÷b÷9f5j4�$²oaHuxw{Z_9u`wyZ]\Pw1��ay�/ª/uxw{ZF�9���bZ}¡y\x��~��SX��9��w{~�ayZ}\=�¸_�­�Z}\`Z]��¡5�«u`wyZ7zU_9uxZ}a?u`~¦�9� �t_b��ux�]_9X¤Zay_9ux��ux~�_ba/ª�_bay�}Z��µzy\`_b¡{��Z]X�~�^Szy\`_bzUZ]\=���æ�G_b\=X�����~�ä}Z]�/ª°�]�9aM¡UZ[¢A�{~�uxZ[zU_;��Z]\O��G�y�r~�aÍ\`Z}®yay~�ay�,�9^`^=�yX¤z�ux~�_9ay^]ª��yZ}\`~�|�~�ay�Mz{\`_b¡1��¡y~���~�ux~�Z]^7_9�o�]_b�ya5u`Z]\=�D���}ux�y�9��^}ª�9ay�[|bZ}\`~��Q�A~�ay����wyZ4uxwyZ}\��]_9ay�]���y^`~�_bay^£�G_b����_;�¼�G\`_9X"zy\`Z}X¤~�^`Z]^t»!�9^£��Zo�yZ]X[_ba��^=u`\x��u`Zo~�a · w1�9z{u`Z]\�÷{��Þ�wyZ©ux\`�9ay^=�¦��u`~�_ba¤�b~�|9Z]av~�a¸cÀÁ{� ì�d;j���cÀÁ�� ì9hbj£^=wy_b�y����wyZ]��z\`Z}^`Z]�9\`��wyZ}\`^ �]_bX2¡{~�ayZou`wyZ�¡¬Z}^=u��GZg��u`�y\`Z}^ _��$u`wyZPut��_¤�9zyzy\=_5�9��wyZ}^]�

�m���m��Î .ç.¨k�®�°k¥�­[¡ ®�¥Á.Ã¥nLO°3­[¡�Â�¡ÄÃ}õÂzâ¡�®�°�w �®>°'¥�­½©X[_bay�¨uxwyZv~�a?|9Z]^eux~��b��ux~�_bay^H�}_bay�y�{�}uxZ}�¾~�aMu`wyZvz¬_�uxZ]a5u`~¦�9� �t_b�{u`�]_bX[Z7�G\x�9X[Z4���_b\`´±ªYuxwyZ�_bayZ��}��_b^=Z]^=u2~�a,^=zy~�\=~�uSux_¸uxw{Z�^=u`\`�y�4ux�y\`�9���9ay�9���A^`~�^H�yZ}^`�}\`~�¡UZ]�¾~�auxw{~�^P��w1�9z�uxZ]\P~�^�êr_9¡y~�ay^}ûÂ^o��_b\=´¸_ba�à=�g���y^x�9�����µ~�a5u`Z]\`z{\`Z}u`Z]�'^=u`\`�y�4ux�y\=Z]��ux\`Z}Z�b\`�9zywy^3á¾cÀê�_b¡y~�ay^�d�÷9fbh{ª�dg÷bfo�bj��«ê�_b¡y~�ay^����9^Hu`wyZ�®1\=^=uHu`_¸\`Z]�9��~�ä]ZFuxwyZ�z¬_��uxZ}a5ux~¦���$_9�¥érZ}�AX¤�9a/ûý^o�]_b�{a?u`Z]\e�D�9�}u`�1�9��a{_9u���u`~�_ba«ùvc jr�9^P���9Z]ayZ}\x�9�$X���u`wyZ4�X¤��ux~��g�9�1���9ay�b�1���bZ��G_b\��]�9�y^`�9�¬~�a{�GZ]\=Z]ay�}Z9ª��9ay��w{Z �y^=Z]�v~�u¥u`_HZ���u`Z]ay�¯ê��y¡y~�a/ûý^cedg÷o�9f5j?àeux~�X¤Z��t~�ay�yZ]zUZ]a{�yZ]a5u¥ux\`Z]��uxX[Z]a5uZá2X¤_A�yZ}�Uu`_H^=u`�y�y~�Z]^���~�uxw��y~�\`Z}�}u���ay�~�a{�y~�\=Z]�}u[Z4­�Z}�}ux^��9ay�¼ux~�X[Z4�J|��9\=�A~�a{��u`\`Z]��uxX[Z]a5ux^}ª��]_9ay�]_bX[~�ux�9a5ux^}ª��9ay�Í_b�{u=��]_9X¤Z}^]�

êr_9¡y~�ay^7�}_bay^=~��yZ}\`Z]�Í�æ^`Z4uTÔ !ÆÞÂÔ|���ÀÀÀi�$Ô�ÒÝߨ_9� u`Z]X[z¬_9\x�9������_b\=�yZ]\=Z]�

Page 56: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d;ì9fyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Ê�y~�^`�]\=Z}u`Z�\`�9ay�y_bX |��9\=~¦�9¡{��Z]^�cD�9^¨~�a×è$~��b�y\=Z'Á{� Ábj¯��ay�º�9^=´9Z}�º�yay�yZ}\¨��w1�Lu�]_9ay�y~�u`~�_ba{^�_ba{Z��g��a�~��yZ}a?u`~��Q�æu`wyZ�Z}­¬Z]�4u2_9���}_ba5ux\=_b��zU_b��~��}� À¼ÿ�ø ! _ba_b�{u`�]_bX[Z]^Æù èÞÔ6Å�ø�ª9��wyZ}\`Z�ø ! Þ�ø ���À0ÀÀi�=ø�ÆAßtèÃÔ×��\`Z�u`wyZ�u`Z]X[z¬_b\`�9�����_b\=�yZ]\=Z]�á�9ay�Üz¬_�uxZ]a5u`~¦�9�����¼X���ay~�zy�{�¦�9¡y��Z¨u`\`Zg�LuxX[Z]a5uv|��9\`~��9¡y��Z}^�_9�S~�a5u`Z]\`Z}^=u]�Þ�wyZS�g�9�{^x�9�°Z4­�Z}�}ur_9��øç! _ba¨ù�����^ Z��{z{\`Z]^=^`Z}�æ�9^�uxwyZ�zy\=_b¡1�9¡{~���~�ut�

�¤c���� Àè! j¥+! ��ÞLù�c j�! �wßs���wyZ}\`Z7uxwyZF�}_b�ya5uxZ}\=�D�9�4ux�1�9�Æ|��9\=~¦�9¡{��Z2ù�c j©^eu��9ay�{^��G_9\ouxwyZ2|��9���yZ2uxw1�Lu�_b�{u=��]_9X¤Zo|��9\=~¦�9¡y��Z]^�ù���_b�y���µux�9´9Z�w1�9�¯uxw{Z�u`\`Z]��uxX[Z]a5u�|��9\`~��9¡y��Z]^�ø ¡¬Z}Z]a �êr_9¡y~�ay^P^=wy_���Z}�Muxwy��u?��c-�.� Àâ! jo~�^P~��yZ}a?u`~�®1Z}�'�G\=_bX&u`wyZF�y~�^eux\=~�¡y�{u`~�_ba�¤c�Ó�j2~��oZ]�9��w+�}_bX¤zU_bayZ}a5u¤ø:¶µ_9�oø ~�^Èà`�9^`^=~��ba{Z]�+�Lu�\`�9ay�y_bXµª��9~�|bZ}aãuxw{Zz1�9^eugªdá'�«ay_9u`~�_ba,Z���zy��~��g��u`Z]���9^S�G_b����_���^]�â��Z4uÈÇ·¶�¡UZ¤u`wyZ�|��9\`~��9¡y��Z]^S_��}�]�y\O�\`~�ay�[¡¬Z4ut��Z}Z]aæø:¶*º>���9a{�«ø:¶�ª±��~�u`wjÇ���¡¬Z}~�ay�[uxwyZ�|��9\`~��9¡y��Z]^rzy\`Z}�]Z]�{~�ay��ø ���å�\`~�u`Z Ç߶õ!Úc�Ç����ÀÀÀi�}Ç·¶gj��bÇ©! ÇQÆ©ªU�9ay� øж[!ÚcGø ���0ÀÀÀ0�=øT¶gj4ªU�9ay�¨�yZ}®yayZøæü� ÇYü� ÔSüPu`_µ¡UZ2~��yZ]a5ux~��g�9�����µä]Z}\`_y��Þ�wyZ7ux\=Zg��u`X¤Z}a?u©øT¶ ! ¶7~�^o^`�9~��¸u`_¡UZ�>L:�:4I8�9K�0`T¤>9B�ß=>9KUTA6LR�8e�9IQà904K«BG.10Ü={>L:4BDª1~���uxw{Z��G_9����_���~�ay�2\`Z}�¦��u`~�_bavw{_b���y^}ÿ

cDù�c j1:G:Pø:¶�� Ç·¶5� øж*º>��! ¶*º>�Oj�À cÀÁ{�Âh � jêr_9¡y~�ay^F�G�y\euxwyZ}\�zy\=_;|9Z]�Ýuxwy��ugª�~��[cÀÁ�� h � j7w{_b���y^[�G_b\[Z}|bZ}\=�¨É¬ª�uxw{Z]a+uxw{Z

�g���y^x�9�°Z4­�Z}�}u ~�^��b~�|bZ}aµ¡5�

�¤c���� Àè! j�!Ì#Ê�Ë��c-�.� ÌÍÆõ� Æ�j

Æå¶ � � ��cÎÌ ¶s� ÌĶ*º>��� ¶Qº>�=j�� cÀÁ{�ÂhbÁ5j

�9a¨Z4��zy\`Z}^`^=~�_ba¸w{ZH�g������Z}�vuxwyZÇàZ±P�â�]_bX[zy�{ux��ux~�_ba¯�9���b_b\`~�uxwyX �G_b\`XH�y�¦�{�dá�Þ�wy~�^Z4��zy\=Z]^=^`~�_9a��g��a[¡UZ��yZ]\=~�|9Z]�[¡?�[��zyzy���A~�ay�P�]_ba{�y~�u`~�_baµcDÁ{� h � j�~�u`Z]\x�Lux~�|9Z]���bª5�9^£~�auxw{ZH�yZ}\`~�|���u`~�_ba¨_9�rcDÁ{� ìLÃ?j4� sâ�Yø ~�^ �yay~�|9��\`~¦�LuxZ9ª1u`wyZ]a¼cDÁ{� h9Á5j�\=Z]�y�{�]Z]^ou`_�uxw{Z^=ux�9ay�1��\`�¨�9�L³=�y^euxX[Z]a5u��G_b\`X2�{�¦�

�¤c���� Àè! jz! # Ê & �¤c���� �}Ì��tj3�¤c�Ì-�=j��z1�9\`�9����Z]��~�ay�ocÀÁ{� ì�Ã?j��Â�°~�´�Z}��~�^`Z9ªL~�a�uxw{Z¥^=z¬Z}�]~��9�5^=u`\`�y�4ux�y\=Z�_9�{è�~��9�y\`Z£Á{�ÂÁ{ª{cÀÁ{�ÂhbÁ5j\`Z}�y�y�}Z]^�ux_«cÀÁ{��d�f5j��

Þ$_¨zy�¦���]Z7uxwy~�^�\`Z]^=�y��u�~�aæu`wyZ¤�}_ba5uxZ���uH_9��_9�y\S�9a1�9����^=~�^�~�a�uxw{~�^���w1�9z{u`Z]\}ª��Z©ay_9u`Zru`w1��u£uxw{Z©�]���9^`^¥_9��^`Z]X[~��O§«�9\=´9_�|A~¦�9a[X¤_A�yZ}��^£^x��u`~�^e�Q��~�ay���9^`^=�yX¤z�ux~�_9a

Page 57: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ�~5ÎãÒoÓeÐ�ÄPÏ�ÐyÐ�Ó4Ù�Ö d;ì�÷cÀÁ�� h � jY�}_b\`\=Z]^=z¬_ba{�y^�ux_��]_bX[zy��Z}uxZ1²�½1��^¥~�a[��wy~���w��9���±��\`\`_���wyZ]�9�y^�zU_b~�a?u`~�ay�ux_¯ø:¶F_b\`~��b~�a1�LuxZ2�G\`_9X�_b¡y^=Z]\=|9Z]�,|��9\=~¦�9¡y��Z]^}�¤stay�{Z]Z]��ªÆ~�a�^`�{��w,X¤_A�yZ}��^]ª�uxw{Zz1�9\=Z]a5ux^��^fO¶1! Ç·¶;� øp¶*º>�Y_���|��9\`~��9¡y��Z�ø:¶�^x�Lux~�^e�Q�7uxwyZr¡1�9��´b�â�y_�_9\��}_bay�y~�ux~�_9a_9�²oZ}®1a{~�ux~�_ba¯Á{�ÂÁ{��dbª

cGø:¶�:<:�ù��<�gf/¶]j1> ? ³ ���wy~���wv~�X[zy��~�Z]^©cÀÁ{�Âh � j�� � ÷[Þ�wy~�^��]�¦��^`^�_9��X[_A�yZ]��^��D�9����^¥�yay�yZ}\�Þ�wyZ]_b\=Z]X�Á{� � � ì�ª��wy~���wµ^eu���u`Z]^�uxwy��ur�9���U�g���y^x�9�°Z4­�Z}�}u`^r~�a�uxwy~�^��]���9^`^��9\=Zo~��{Z]a5ux~�®±�9¡y��ZP�9ay�µ�9\=Z�b~�|bZ]aµ¡5�vuxwyZPu`\`�yay�]��uxZ}�¨�D�9�4ux_b\=~�ä]��ux~�_ba��G_b\=X2�y�¦�2_���cÀÁ���dgÃ5j4Û{u`wyZP�G_b\`X2�{�¦�2�]_L�~�a{�]~��{Z]^Æ��~�u`wµcÀÁ{�ÂhbÁ5jÆ�L�QuxZ]\YX¤�9\=�b~�a1����~�ä}~�ay��_�|bZ}\YuxwyZ��yay�}_ba5ux\=_b����Z]�7�]_�|��9\=~¦��u`Z]^]�

Þ�wyZ�^eux\`�{�}ux�{\x�9���9a1�9���A^`~�^�~�a5u`\`_A�y�y�]Z}�H~�a�uxw{~�^���w1�9z{u`Z]\�^=�yzyzU_b\=u`^Y�9ay���bZ]aA�Z]\`�9��~�ä]Z}^$ê�_b¡y~�ay^]ûý^�\`Z}^`�y��u��G\=_bX���ayZ}�,u`wyZ]_9\`Z}u`~��]�9�5z¬Z}\`^=z¬Z}�}ux~�|bZ��Yè�~�\=^=u]ªL_9a�uxw{ZuxZ}��wyay~��g�9���G\`_ba5u]ªUu`wy~�^ �9a1�9���A^`~�^r_9­¬Z]\=^©^=�A^euxZ]X¤��u`~������g�A^o_9�YX¤�9a1�9�9~�ay�[X¤_A���Z]��^H��~�u`w��yayX[Zg�9^=�y\`Z}�¾�]_ba{�G_9�yay�yZ}\`^µcD~À�ÂZ���ªY�{ay_b¡y^=Z]\=|9Z]�Íz1�9\`Z}a5ux^7_9�r�]_9a?u`\`_b�|��9\`~��9¡y��Z]^]ª£��^7~�a¾è�~��b�y\`Z}^7Á{�Âf1cD�1jV��cG�?j`j�ªÆ��wyZ]\=Z¨ê�_b¡y~�ay^]ûý^2^eu��9\eux~�a{���9^`^=�yX¤zA�ux~�_baMcÀÁ{�Âh � j�~�^r~�a1�9zyz{��~��]�9¡y��Z9��¿�Z]�}_bay�/ªU_baµuxwyZS�]_ba{�]Z]z�ux�1�9���G\`_9a?u]ª1uxwyZS^=u`\`�y���ux�{\x�9���G\x��X¤Z4��_b\=´�\`Z}zy\`Z}^`Z]a5u`^¯�'�G�yay�y�9X¤Z}a5u��9��^`w{~��Qu[�G\`_bX u`wyZ¨|9_��]�9¡y�y���9\=�_9�°�}_b�ya5uxZ}\=�D�9�4ux�1����~�a{�yZ]zUZ]ay�{Z]ay�}~�Z]^ocDZ��Â�y��cÀÁ�� h � j`jÆux_�u`wyZ |9_��]�9¡y�y���9\=�[_9�/zy\`_���]Z}^`^=Z]^¥�9ay�2X[Z]��wy�9ay~�^=X¤^}ª9�G\`_9X���w{~���w2w?�yX¤�9a ³=�y�y�9X¤Z}a?u�_9�±�]_9�ya5uxZ]\e�D�9�}u`�1�9��^_b\=~��b~�a1��uxZ}^]��½©��u`wy_b�y�9w¨Z��{z{\`Z]^=^`~�_bay^o_9�Æ�]_b�ya5u`Z]\=�D���}ux�y�9��~�ay�yZ}z¬Z}ay�yZ}ay�]~�Z]^o�]�9a¡UZ�Z}ay�b~�a{Z]Z]\=Z]�«u`_¤�D�9�}~���~�u��LuxZP�9���bZ]¡{\x�9~��P�yZ]\=~�|���ux~�_bay^�_9�Y�]�9�y^`�9��Z}­¬Z]�}u`^FcÀ��^ ~�acÀÁ�� ì�Ã5j`j4ªb�9\=u`~��]�{�¦��u`~�ay�ruxwyZ�\=~��bw5uY~�ay�yZ]zUZ]a{�yZ]ay�}~�Z}^£�G_9\Y�Pzy\`_b¡{��Z]X_b\£�9^=^`Z}^`^`~�ay�uxw{Z��9^`^=�yX¤z�ux~�_9ay^�¡¬Z}wy~�a{�v^`�y��wµ~�ay�{Z]zUZ]ay�yZ}ay�]~�Z]^�X¤����_9�QuxZ}av¡¬Zou`wyZPw1�9\=�yZ]^euz1�9\euH_���uxw{Z¤zy\=_b¡y��Z]Xµ��sta�u`wyZ�^eux\=�y�}u`�y\x�9�Y�G\`�9X¤Z4��_9\`´±ª$u`wyZ¤�}_b�ya5uxZ}\=�D�9�4ux�1���Z4��zy\=Z]^=^`~�_9ay^µu`wyZ]X[^`Z}��|9Z]^µ�9\`Z��yZ}\`~�|bZ]��cD~��SayZ]Z}�yZ]�±j��G\=_bX ��X¤��uxwyZ}X��Lux~��]�9�uxw{Z]_b\e�æcD�9^�~�aæcÀÁ�� ì9hbj��9ay�æcÀÁ{�Âh{d;j=j4�£¿Au`~����Jª?ê�_b¡y~�a{^]ûý^�zy~�_bayZ]Z}\`~�ay�H\=Z]^`Z]�9\`��w¯w1��^zy\=_;|9Z]aÜcG~Qj u`w1��u��9���9Z]¡y\`�9~��7X[Z}u`wy_��{^S�]�9aMwy�9ay�y��Z[�]�9�y^`�9�¥�9a1�9����^=~�^P~�a��}_bXF�zy��Z4�¨X2�y��ux~�^eu��9�9Z�zy\=_b¡y��Z]X[^©�9a{�¼cD~�~¦j�uxw1��u©�g�9�{^x�9�$Z}­¬Z]�}u`^o~�a«^`�y��w�zy\=_b¡y��Z}X¤^�g��a[¡UZ�\`Z}�y�y�}Z]�¤u`_PZ]^eux~�X¤�9¡y��Z�¢?�1�9a5ux~�ux~�Z]^rcD^=Z]Zr�9��^`_S¿�Z}�}ux~�_bay^£Á{�Âh{��d��9ay�FÃy�ýÃ?j4�

B�ÿ3��y\J4F3a4H_K�J4LMx�F3y\�i~sQ8�G{ �i÷iT�f3K�Hõv5F�Iiv2J�K�L\H_F3YAv�1K�`G`GyM�L\HGh�J�DGFÂJ4agK�H_W4yMK�J4LMI�Hja4XGyMF�I�lEQ8�G{<ò$�0TJ4I�hia�K�`GD_W�UVL\J4DmHGI�f'IiH2lSI�X_H_Y2LMHGh�K�agfkW�v;F'J-U|FkF3H���ÏeK�H_Y °�ø�±�N*¦¼Ïn²i{���I�J4Fi~�DGI:U|FkxiFkaZ~�J4D_K�JJ4D_FAL\}�`GyML)fkK:J�L\IiH·h�IF3W�I�HGyM�?IiHGFAUEK��;c�üVIivGL\H5W#� WzfkI�H_Y2L\J4LMI�H L)W�J�DGFAU|F3K$PiF3W�JzK�W�W�X_}�`GJ4LMI�HHGF3F3Y2FZY1lSIiaVLMYGFkH0J4L\lS�L\HGh�J�DGF�fkK�X_W�K�y�F¹��F3f'J3{

Page 58: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

d�hbiyÄÆÅ ÇoÈ©É$Ê£ËÍÌAΫÄ�Ç[Ï�Ð�Ç©Ñ�ÒoÓtÇSÔrË�ÇPÕ«ÐPÇ©Ö©Ò×É$ũʨÓOÒPÊ¥Ö7É�ÓOØ�Ó4Ä�Ç©É�Ó4Ù�Ö!Ù�Ø�Ä�Ç[Ï$Ð�ÇoÑ�Êk×óÝ� ò �Æqrl ö h òÞ�wyZ���_9\`´v\`Z}�]_b�{a?u`Z]�¨~�avuxwy~�^���w1�9z{u`Z]\r^`zy\`�9ay�F�G\`_bX ut��_¤^`~�X[zy��Zo~��yZ]�9^�uxw1�Luux_�u��9�����¸��wy�9ay�bZ}�,XH����u=ux~�ux�y�yZFux_����9\`���g�9�{^x�9��~�ut�b�¤Þ�wyZ¤®y\`^=uH~��yZ]�¨��\`_b^=Z¤~�auxw{Zo^`�yX[X[Z]\�_��£d�÷b÷bi{ª?��wy~���Zrs£���9^���_b\=´A~�ay�H��~�uxw�Þ�_bX µ£Z]\`X¤�H_baÕà=½×Þ�wyZ��_b\e�7_9�¬sta{�GZ]\=\`Z]� · �9�y^`��ux~�_ba�á¤c4´�Zg�9\=�±�9ay��µ£Z]\`X¤�7d�÷b÷{d9ÛA^=Z]Z �9��^=_ · w1�9z{u`Z]\ � j4�å�Z¨zy���g�bZ]�ã�9\`_9�yay�¾��~�uxw¾uxw{ZµzU_b^=^`~�¡{~���~�ut�,_9� \`Z}zy�¦���]~�a{��u`wyZµz1�9\`Z}a5ux^e�â��wy~����\`Z}�¦��u`~�_ba{^`wy~�z �¤c >º � ¼w½ º jP��~�uxw�~�ux^S�G�yay�}u`~�_bay�9���}_b�ya5uxZ}\`z1��\=u >º !Ö» º c%¼w½ º ��Ô º j�9ay��ª�^`�y�{�yZ]ay���bª�Z4|bZ]\e�?uxwy~�ay�,¡UZ]�5�9a¼ux_'�D�9����~�a?u`_�z{�¦�9�}Z9ÿ¨��Z¨®ya1�9������w1�9�ã�X¤��uxwyZ}X��Lux~��]�9��_b¡?³=Z]�}u¤ux_���wy~���wÍ��Z¨�]_b�y���+��u=ux\=~�¡y�{u`Zv�D�9X¤~���~��9\Hzy\=_bzUZ]\=u`~�Z}^_9��zyw5�A^`~��g�9��X¤Z}��w1�9ay~�^`X[^H~�ay^=u`Zg�9�,_���uxwy_b^=Z�^`��~�zyzUZ]\e�'Z]zy~�^=uxZ}X¤~��¤z{\`_b¡1��¡y~���~��ux~�Z]^��¤c >º � ¼w½ º j���~�uxw,��wy~���w¼��Z¯w1�9�¾¡UZ]Z]a¾��_b\`´?~�a{�'^=_���_bay�æ~�a,uxwyZ¯^eux�y�{�_9�?ã��g�9Z]^`~��9aÝayZ4ut��_b\=´?^]� ²P�9aya5� ��Z}~��bZ}\]ª���wy_����9^���\=~�u`~�ay�'wy~�^¤�y~�^=^`Z}\=u��;�ux~�_ba¯��u�u`w1��u�u`~�X[Z9ª{�9^`´�Z]�¯��~�uxwµ�9^eux_ba{~�^`w{X¤Z}a?u]ÿ?àZ²oZ}u`Z]\=X¤~�ay~�^eux~��rZ]¢?�1�Lux~�_9ay^9ðÞ�\`�y���v�yZ}u`Z]\=X¤~�ay~�^eux~�� ðá'½ ��u`wy_b�y�bwv��Z�´Aa{Z}�þuxwy��u��yZ}u`Z]\=X¤~�ay~�^eux~��©^=ux\=�y�}u`�y\x���Z]¢?�1�Lux~�_9ay^£w1�g|bZo����_9ay��wy~�^eux_b\e�7~�aFZ]�}_bay_bX[Z}u`\`~��]^]ª5��Zr|A~�Z}��Z]��u`wy~�^£\`Z}zy\`Z}^`Z}a��u��Lux~�_9a��9^S�µ\=Z]��~��7_9�¥uxwyZFz1�9^eug�¤è1_b\P�y^S��uå · �°½Ú~�aæuxw{Z[Z]�9\`����d�÷b÷9i9^]ª°uxw{Z~��{Zg��_9�Æzy�{u`u`~�ay�¤uxwyZ�^`Z]X¤�9a5ux~��]^o_���ã��g�bZ}^`~¦��aæayZ}ut��_b\`´?^�_baæ�¤�yZ}u`Z]\`X[~�a{~�^=u`~���G_b�ya{�1��ux~�_bav^`Z}Z]X[Z]�«�Fw{Z]\`Z}^=�µ_9��uxwyZP��_b\`^eur´A~�ay�/�

Þ�wyZo^=Z]�]_9ay�v^`~�X¤zy��Z ~��yZg�S�g�9X[Z��G\`_bX¹´$Z4uxZ}\�¿�zy~�\=uxZ}^]ûý^���Z]�}u`�y\`Z©��u�u`wyZ©staA�uxZ}\`a1�Lux~�_9a1�9� · _bay�b\=Z]^=^Y_9��´Ywy~���_b^`_bz{w?�S_9�¬¿A�]~�Z}ay�]ZSc�å zyz{^x�9���{ª5¿A��Z]�{Z]a/ª±d�÷b÷�d;j4�staH_9ayZ£_9�ywy~�^�^`��~��yZ]^}ª;´�Z}u`Z]\�~������y^=u`\x��u`Z]��wy_��¾���g�9�{^x�9�?�y~��9�b\x��Xá��_b�y���S��w1�9a{�bZ��wyZ}aM��|��9\=~¦�9¡y��Z�~�^oX¤�9ay~�z{�y�¦��u`Z]�/�PÞ�_¯X¤Z�ª�uxwy��uP^`��~��yZH_9��¿�zy~�\=uxZ}^]ûý^O»+��wyZ]a�]_9X2¡y~�a{Z]�¨��~�uxw«u`wyZH�{Z}uxZ}\`X[~�ay~�^=u`~���^=u`\`�y�4ux�y\`�9�$Z}¢?�1��ux~�_bay^t»Ý���9^©uxw{ZH´9Z4�¨u`_�ya{�G_9���y~�ay�PuxwyZrX���ay~�zy�{�¦��u`~�|bZ����]�]_9�ya5u�_9�/�g�9�y^`��ux~�_ba¤�9ay����Z]�Fux_�X¤_b^eu¥_9�/uxw{ZZ4��zy��_b\x�Lux~�_9ay^��yZ]^=�]\=~�¡UZ]�µ~�avuxwy~�^���w1�9z{u`Z]\}�

sH^=wy_b�y���Í\=Zg�9������X¤Z}a?u`~�_ba¼�9a{_9uxwyZ}\F~�ay�}~��yZ}a?u7u`w1��u[�]_9a?u`\`~�¡y�{uxZ}�¼u`_æuxwy~�^��w1�9z�uxZ]\}�7sta�Z]�9\`���Mdg÷b÷bÁvs \=Zg�9�æu`wyZ7®1Z]\=�]ZF�yZ]¡y��uxZF¡UZ}ut��Z]Z]a�½©\euxw?�y\,��_9�����¡UZ]\`�9Z]\���ay�+é �9aya5�Íå�Z]\=X2�{u`w+_baÍu`wyZ«X[Zg��ay~�ay�'_9�P^eux\=�y�}u`�y\x�9� Z]¢?�1��u`~�_ba{^c��P_b���y¡¬Z}\`�bZ}\Æd�÷b÷ � ÛLå�Z]\=X2�{u`w¤d�÷9÷ � j4��sâu�^`�{�y�yZ]a{����wy~�u°X¤ZÆu`w1��u�uxw{Z¥�}Z]a5ux�{\=�5�_b���!u`Z]ay^=~�_ba!¡¬Z4ut��Z}Z]aþZ}�]_bay_9X¤~�^=ux^µ�9a{�×^=ux��ux~�^=u`~��]~��9ay^v^euxZ]X[^¯�G\`_bX ^=~�X[zy��Z^`Z}X���a?u`~��¸�}_ba{�G�y^=~�_ba�ÿ�¿Aux��ux~�^=ux~��]~��9ay^[\`Zg���!^eux\=�y�}u`�y\x�9�©Z]¢?�1��u`~�_bay^¯��^�^=ux��uxZ��X[Z]a5ux^��9¡¬_9�{u-G�cÀùâ� j4ª?��wy~���Z Z}�]_ba{_bX¤~�^=u`^�\=Zg�9��u`wyZ]X��9^DG�cÀù��<êsë�c j`j��ÆÞ�wy~�^��_b�y���FZ��{z{�¦�9~�aH��w5�7^=ux��ux~�^=ux~��]~��9ay^��}�¦�9~�Xu`w1��uY^eux\=�y�}u`�y\x�9�{Z]¢?�1��u`~�_bay^Ywy��|9Z�ay_X[Zg�9ay~�ay�F�9ay�v��w5�¯Z]�}_bay_bX[~�^eux^�\=Z}u`_b\=u�u`w1��u�^=ux��ux~�^=u`~��]^�wy�9^�ay_[^`�y¡y^eu��9a{�]Z9�¥s��\`_�uxZv�«u`Z]��wyay~��g�9��\`Z}z¬_b\eugª?àx²oaMu`wyZ¯¿Aux��ux~�^=u`~��g����sta5uxZ}\`zy\=Z}ux��ux~�_ba�_9� ¿Au`\`�y���ux�{\x�9�$WY¢A�y��ux~�_bay^Zá'c�´$Z]�9\`��d�÷b÷bÁ9�gj4ªUwy_bzy~�ay��u`_�^`Z}ZHuxwyZ�ut��_v�]�9X¤z{^©Z]XH¡y\x�9�}Z

Page 59: òYö - University of California, Los Angelesbayes.cs.ucla.edu/BOOK-99/ch3.pdfux~ _ba¾db ÂÁ5j4ª ^` y wzy\`Z} y~ 4ux~ _bay^S 9\`ZFuxw{Z[wy 9 X 9\=´¨_ g y^x 9 £X[_ {Z] ~ ay yª

ÌAÎ�~5ÎãÒoÓeÐ�ÄPÏ�ÐyÐ�Ó4Ù�Ö d�h�d~�a�\`Z}�]_bay�}~���~¦��u`~�_ba/�¯ér_9uxw{~�ay�¨_9��uxw{Z�^`_b\eu2w1��zyz¬Z}ayZ]���«Þ�wyZ�^eu���u`~�^eux~��}~¦�9a{^�~�auxw{Zo�y~�^=zy�{u`ZP�]_ba5ux~�a?�yZ]��u`_7~�ay^=~�^=u�uxw1�Lu��9a5�?uxwy~�ay�Huxwy��u�~�^�ay_9u�~�a5uxZ]\=zy\`Z4uxZ}�¨��^G�cDù¬� j7^=~�X[zy�����¦����´A^7X[Zg��ay~�ay�{�,Þ�wyZ¯Z]�}_bay_bX[~�^eux^]ª¥~�a��}_ba5ux\`�9^=u]ª��9\=Zv^=ux~����ux\e�A~�ay�2u`_[�{Z]�]~��yZP~���~�u�����^Aê�ë�c j�u`w1��u�u`wyZ}��w1�g|bZS¡¬Z}Z]a¯X¤Z]�9ay~�ay�2ux_7^`���v�9����9��_bay�y�

WYay�]_b�{\x�9�b~�ay�v�]_9����Z]�9�b�yZ}^©\=Z]�]Z}~�|9Z2�D�9\©ux_A_���~�u`u`��Z��]\=Z]�y~�uP~�a«_9¶[�]~¦������w1�9aA�ayZ}��^]ª��]_bay^=~��yZ}\`~�ay��uxw{Z7~�X[X¤Z}ay^`Z�~�X[z1�9�}u uxwyZ4�«wy��|9ZF_ba¸uxwyZHZ]ay�}_b�y\x���bZ]�/�SsX2�y^eu�u���´9Z«u`wy~�^¤_bzyzU_b\=u`�yay~�ut�¼ux_,�9��´?ay_�����Z]�{�bZ¸�G_b�y\¤�]_b����Zg���b�yZ]^[��wy_�^x�g��]���9\`~�ut��^=wy~�ay~�ay�2uxw{\`_b�y�9wµu`wyZ?ê�ë�c j�_bzUZ]\`��ux_9\�¡¬Z4�G_b\`ZP~�u��5�9~�a{Z]�¯z¬_9zy�y�¦��\`~�ut�Uÿ¿Au`Z}­¬Z]aÄ���9�y\=~�uxä}Z]a/ª^²P�g|A~��¸è1\=Z]Z]�{X��9a�ª\Ðb�9X[Z]^©êr_9¡y~�ay^}ª���ay� ´Ywy~���~�zâ²P�g��~��/�´Ywy~��¬^`wy_���Z]�µ^`zUZ]�}~¦�9�¬�]_b�y\`�9�bZo~�a�zy\=~�a5ux~�ay�HXH�¤z1�9zUZ]\�~�aÒÑ�IJ6LR�04BDß�IqÛ9>2c4´�Zg�9\=�d�÷b÷bì9�5j4ª�uxwyZ�³=_b�y\=a1�9�{�G_9�yay�yZ}�¤¡5�7�]�9�y^x����~�ut�Uûý^Æ��_9\`^=u��9�{|9Z]\=^x�9\e�5»�u��9\`�w´�Zg�9\O�^`_9a/�