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N3,1 N3,2 N3,3 · · · N3,l−2888 888Nl−1,1 Nl−1,2
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DG@IFQ=%�nR�B AJA@IBEBQR�L�DzBQRUFE�~=%PC@IBQR�F >x=%LAB�JAL���DG=vDAi ∈ {A1, . . . ,Al}
=~£%R�ZTJALART AFQKnHA=%HA@;wPI@CDQ�
pi
&�=�£%R�Z0 < pi < 1
&i = 1, . . . ,l
&�R2D ∑l
i=1 pi = 18��/@
Ni
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Ai
PCKnFQB�>AR�BnDG@CFG=%�nR�B'&(=%PCKnFQB�KnL�>A@CD���JARtPCR £nR�Z2DGR�JAF =%PC�~=3w
DGKn@CFQR −→N = (N1, . . . ,Nl)
(KnBQBEr�>AR JjLARs>A@CBEDGFE@IHAJ9DG@IK%L0MOJjPCDG@CLAKnMq@u=%PCR >ARs A=%FG=%Mqr�DQFQR�B(n,p1, . . . ,pl)
8<5= �fKnLAZ2DG@IK%LN>ARS jFQKnHx=vHA@IPC@CDG�z>ARUZ�R £nR�Z�DQR�JAFz=%PI��=vDGK%@IFQRYB'#¢��Z�FE@CD',
P (N1 = n1, . . . ,Nl = nl) =n!
n1!n2! · · ·nl!
l∏
i=1
pini
+ Kn@CZ�@ =%PIKnFEB >A@ ����FER�LmDGB�MqKnMqR�LmDGB >ARUZ�R�DQDQRS>j@IBEDQFQ@CHAJjDG@CKnL ,
E(Ni) = npi
V ar(Ni) = npi(1− pi)
Cov(Ni,Nj) = n(−pipj)&x (KnJAF
i 6= j# L� �R�JjDz=%JABEBQ@5Z~=%PCZ�JAPCR�F PI= �fKnLAZ�DQ@IKnLN�%��LA��FG=vDQFQ@IZ�RU>AR�B�MqKnM0R�LmDGB',
V
M−→N
(−→t ) = E(e
−→t ·
−→N ) = E(et1N1+···+tlNl) = E(
l∏
i=1
etiNi)
=∑
n1,...,nlt.q.∑
i ni=n
n!
n1! · · ·nl!p1
n1 · · · plnl
l∏
i=1
etini
=∑
n1,...,nlt.q.∑
i ni=n
n!
n1! · · ·nl!
l∏
i=1
(pieti)ni
= (∑
i
pieti)n
∑
n1,...,nlt.q.∑
i ni=n
n!
n1! · · ·nl!
l∏
i=1
(pie
ti
∑
i pieti)ni
= (∑
i
pieti)n
= (p1et1 + · · ·+ ple
tl)n
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Xi
&i = 1, . . . ,l
BEKnLmDHA@ILjKnM0@I=%PIR�B
(n,pi)8
798 # L�MqR�LmDG@CKnLALAR Z�R�DEDGRU AFQK% AFQ@C��DG� BEKnJAB9�fKnFQMqRs>ART AFEKn �K%BQ@CDQ@IKnL0,�S���¦c��(.~&���&'�(� � ������� −→
N = (N1, . . . ,Nl) ∼ Multinomiale(n,p1, . . . ,pl)� ������� Mk = N1 + · · ·+ Nk � ��� ����������������� � Mk ∼ Binomiale(n,p1 +· · ·+ pk) ��S�������# L JjDQ@IPI@CBQR PI= �fKnLjZ�DG@CKnLq�n��LA��FG=vDQFQ@CZ�R�>AR�ByM0KnMqR�LmDQByZ�=%PIZ�JAPI��R� jFQ��Z���>AR�M0MqR�LmD�8
MMk(t) = E(etMk) = E(et(N1+···+Nk))
= E(et(N1+···+Nk)+o(Nk+1+···+Nl))
= (p1et + · · ·+ pke
t + pk+1e0 + · · ·+ ple
0)n
= ((p1 + · · ·+ pk)et + (pk+1 + · · ·+ pl)e
0)n
= (pet + (1− p))n &AK��p = p1 + · · ·+ pk
� R �/Jj@ R�BED �EJjBEDGR�M0R�LmD�PI= �fKnLAZ2DG@IK%L �n��LA��FG=vDQFQ@CZ�R >AR�B�MqKnMqR�LmDGB > #�JjLAR PIKn@HA@ILjKnM0@I=%PIRz>ARU x=%FQ=%M0r2DGFER�B �'L0&� � 8
2
�
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M0R�L�DQ@IKnLjLAR�PCPIRs>ARUPI=O>A@IB DGFQ@CHAJjDQ@IKnL � �2D\=j8 � PCPIRsR�BEDz>A���ALA@IR >ARUPI=OM =%Lj@Ir�FER BQJA@;£v=%LmDGR�,� L�£nR�Z2DGR�JAF =vPI�~=3DGKn@CFQR −→
Θ = (Θ1, . . . ,Θl) (KnBQBEr�>AR�JALjRN>A@IB DGFQ@CHAJjDQ@IKnL >AR�XY@;w
FQ@IZ\[jPIR�DS>AR0 x=%FG=vM0r2DGFQR�B(α1, . . . ,αl)
PIK%FQBQ��JAR0Z\[x=%��JAR £v=%FE@u=%HjPIRθi
R�BEDO>j���xLA@CR x=vFθi = Zi/
∑lj=1 Zj
=�£%R�Z PIR�B�£�8�=j8Zj
DQR�PIPCR�B ��JARZj ∼ Gamma(αj ,1)
&~ (KnJAF!���O:*&;8C8;8 &¢P 8# L�= =%PIK%FQB ��JAR ∑l
j=1 Θj = 18
<5= �fKnLAZ2DG@IK%LN>ARS>jR�LABE@CDG�U>ARUZ�RU£%R�Z2DGR�JjFY=%PC�~=vDQKn@IFERsB'#¢��Z�FQ@;D�=%@ILABE@%,
f(θ1, . . . ,θl) =Γ(α1 + · · ·+ αl)
Γ(α1) · · ·Γ(αl)
l∏
j=1
θαi−1j θj ≥ 0
# L{ (R�J9D?=%@CBQ��MqR�LmDSZ~=%PCZ�JAPCR�FT>A@ ����FER�LmDGBSM0K%M0R�L�DQBS>jR Z�R2DQDQR >A@IB DGFE@IHAJjDQ@IKnL �fK �α =
∑nj=1 αj
&xR2Di 6= k
�-,E(Θk) =
αk
α
E(Θ2k) =
αk(αk + 1)
α(α + 1)
E(ΘiΘk) =αkαi
α(α + 1)
E(Θk(1−Θk)) =αk(α− αk)
α(α + 1)
V ar(Θk) =αk(α− αk)
α2(α + 1)
Cov(Θi,Θk) = −αkαi
α2(α + 1)1 KnJAF PI= BEJA@CDQR�& @IP�R�BED @ILmDGR�FQR�BEBG=%LmDN>jRbMqR�LmDG@CKnLALAR�F =%JABEBQ@zPIR�B0>AR�J �FQ=% A (KnFEDQBBQJA@;£%=vL�DQB��/Jj@5£v=%PCR�LmD
α,
α =E(Θk(1− Θk))
V ar(Θk)
α = −E(ΘiΘk)
Cov(Θi,Θk)# L�=0=vPIKnFEB PIR�B AFQKn jFQ@I�2DG��B�BEJA@C£v=%LmDQR�B',:n8�< KnFQBE�/JjRSP��T7 &jKnL�=?�/JjR
(Θ1,Θ2 = 1− Θ1) ∼ Beta(α1,α2)
]
798 # L�MqR�LmDG@CKnLALAR Z�R�DEDGRU AFQK% AFQ@C��DG� BEKnJAB9�fKnFQMqRs>ART AFEKn �K%BQ@CDQ@IKnL0,�S���¦c��(.~&���&'�(��* ���
(Θ1, . . . ,Θl) ∼ Dirichlet(α1, . . . ,αl)� � � � � γ1, . . . ,γl���� � � � � � � � � � ��� � � � � ��� � 0 < γ1 < γ2 < . . . < γn = l
� � � ����� ��� � ��� �
(
γ1∑
j=1
Θj,
γ2∑
j=γ1+1
Θj, . . . ,
γn∑
j=γn−1+1
Θj) ∼ Dirichlet(
γ1∑
j=1
αj,
γ2∑
j=γ1+1
αj, . . . ,
γn∑
j=γn−1+1
αj)
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2
�
� � �����( ���� �
� ������ �� � ����� � � �� � � �� ~Θ
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N1,1 N1,2 N1,3 · · · N1,l
N2,1 N2,2 N2,3 · · · N2,l−1
N3,1 N3,2 N3,3 · · · N3,l−2888 888Nl−1,1 Nl−1,2
Nl,1# L�BEJA A (KnBQR ��JAR �KnJjFtZ\[x=%��JAR =%LALA��R >AR BE@ILA@CBEDQFQR PCR0MqKnLmD\=vL�DSDGK%DG=%P->AR�BtBQ@;wLA@IB DGFQR�B�R�BED�Z�KnLAL�JAB �'��JAR P%#¢KnL LAK%DQR�FG=vB
Ni
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�zKvDGKnLAB���JARNj =
l∑
i=1
Nij, ∀j
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Θkj=�£%R�Zk = 1, . . . ,l
(KnJAF�JAL � �6 9�/�28�< R�B� AFQ��>A@IZ2DG@CKnLAB�>AKn@C£%R�LmDzBQR �'=%@IFERTR�L"�fKnLjZ�DG@CKnL>AJ�LjKnMOHAFER�B�>ARUBQ@CLA@IB DGFQR�B�>A� � 4?>A��Z�Pu=vFQ��B�PCKnFQB >AR�BY=vLALA��R�B� AFQ��Z���>jR�LmDGR�B�8
# L�Z\[AR�FEZ\[ARt>jKnLAZ 4?Z~=%PCZ�JAPCR�F',
p
E(Θkj|N1j = n1j ,N2j = n2j , · · · ,N(k−1)j = n(k−1)j)��K%DGFERSMqK9>jr�PIRULARUDGR�Lx=%LmDsZ�KnMq jDGRT��JARt> #¢JALARt=%LALA��Rt>ARSBQ@CLA@IB DGFQR8&xLAKnJABY=%PIPCKnLAB
>AKnLAZ8&9 x=vF BEKnJAZ�@IB >ARzZ�Pu=%F DG�8&mKnM0R2DQDQFQR�>ARzMqR�LmDQ@IKnLALjR�F-P%#�=%LALA��Rz>jRzBE@ILA@CBEDGFER�>x=%LAB PIR�B@ILA>A@CZ�R�B �©@CR ,
Nj ≡R�D
Nij ≡ Ni
� 8
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LA@IB DGFQR�B�BQJAF £nR�L�JAR�Bs x=%F�=vLALA��R�B�>ARU>A��£%R�PIK% A �R�M0R�LmD ~NR�BED�Z�K%LAL�JARSBE@ P%#¢KnLNZ�K%LALx=%@;D
PIRTLA@C£%R~=%Jd>jRSFE@IBQ��JAR ~Θ>ARSP%#�=%LALA��RS>ARSBQ@CLA@IB DGFQRv8 � R�DQDQRt>A@CBEDQFQ@IHjJjDG@CKnL�R�BEDYBQJA A (KnBE��R
MOJAPCDQ@ILAK%M0@I=%PIR�>ARU x=%FQ=%Mqr�DGFER�B�,
~N |~Θ ∼ Multinomial(N,Θ1,Θ2, . . . ,Θk)X #�=%JjDQFQR� x=%F D(&�LAKnJjB =%PCPIKnLjBOBQJA j �KnBER�F0�/JjR�PCR�Lj@C£nR�=%Jg>AR�FE@IBE�/JjR ~Θ
BQJA@;D0JjLAR>A@IB DGFQ@CHAJjDQ@IKnLN>ARTXY@CFQ@CZ\[APIR2D(,
~Θ ∼ Dirichlet(α1,α2, . . . ,αk)
< R�B �fKnLjZ�DG@CKnLAB >AR FE�� x=%F DG@;DG@IK%L >AR ~N |~ΘR�D ~Θ
BEKnLmDSLjKnM0Mq��R?FER�BE �R�Z�DG@;£nR�M0R�L�D'&f(~n; ~θ)
R2Du(~θ)
8� KnF D$>jR Z�R�B©[ � �KvDG[Ar�BQR�B�&nLAKnJjB©=%PCPIKnLjB">j��DGR�FQMq@ILAR�F$PI=�>j@IBEDQFQ@CHAJjDG@CKnL 4z (KnBEDQ��FE@IKnFE@%&Z�#¢R�BED.4?>A@IFER Pu=?>A@CBEDGFE@IHAJ9DG@IK%L >AR ~Θ| ~N
81 KnJAF�Z�R�Pu=6&AK%L�J9DG@IPC@IBERsPu= �fKnFEMOJAPCR >AR � = � R�B���JA@ LAKnJjB�>jKnLALAR8,
u(~Θ| ~N) =u(~θ)f(~n; ~θ)
∫
u(~θ)f(~n; ~θ)dθ
∝ u(~θ)f(~n; ~θ)
∝l
∏
i=1
θαi−1i
l∏
i=1
θn1
i
∝
l∏
i=1
θαi+ni−1i
# L�FQR�M =%FE��JARU�/JjRu(~Θ| ~N)
R�BED�JALAR �fKnLAZ2DG@IK%LN>ARUFQ�� A=%FEDQ@CDG@CKnLN> #¢JALART>A@IB DGFE@IHAJ9wDG@IK%L�>jRSXz@IFQ@CZ\[APIR2D >ARU x=%FQ=%Mqr�DGFER�B
α1,α2, . . . ,αl
DGR�PCB ��JAR�,
:~}
α1 = α1 + N1,α2 = α2 + N2,
. . . ,αl = αl + Nl# L�LAK%DQRT=%JABQBE@%,α =
∑l
i=1 αi# L�KnHjDQ@IR�LmD�>AK%LAZ�,
u(~Θ| ~N) =Γ(α)
∏l
i=1 Γ(αi)θα1−11 . . . θαl−1
l
� R2DQDGRY AFQKn jFQ@I�2DG�z AFQKn jFQR 4tPI=SZ�KnL��EKnLAZ2DG@CKnL >AR�B >A@IB DGFE@IHAJjDQ@IKnLjB MtJAPCDQ@ILAKnMq@u=vPIR R2DXY@IFE@IZ\[APCR�DzLAKnJABz �R�FQMqR�Dz>AROFQR2DGFEKnJj£nR�FYJALjRO>A@IB DGFE@IHAJjDQ@IJAK%L 4q �KnB DG��FQ@IK%FQ@5BQJA@;£v=%LmDYPI=PIKn@�>ARUXY@IFE@IZ\[APCR�D�8A��R�M0=%FQ��JAKnLjB ��JARULAKnJAB�=~£nK%LAB�Z\[AKn@IBE@�Z�R�B�>AR�J! NPIKn@CB >x=%LjB PIRsHAJ9D>ARUH���LA���xZ�@IR�Fz>ARUZ�R�DQDQRU AFQKn jFQ@I�2DG�v8
# L� �R�JjDz=%@ILjBQ@ =%@CBQ��M0R�L�D�>j��DGR�FQMq@ILAR�FE(~Θ| ~N)
,
E(~θk| ~N) =αk
α, ∀k = 1,2, . . . ,l
=αk + Nk
α + N
� � � � " � �� � ��" � ������������� ��� �b=v@ILmDGR�Lx=%LmD(&ALjKnJAB LAKnJjB @ILmDG��FQR�BEBQKnLjBz=vJNM0K/>Ar�PCRY APCJAB �n��LA��FQ=%P%&9K � ��R�B D DGKnJ9w
�EKnJAFQBU>AK%LALA��& M0=%@CBY x=%BUPI= >A@IB DGFQ@CHAJjDQ@IKnLd>jR ~Θ8 � #�= � =%LmDt A=%B > #¢[ � (K%DG[jr�BQR0BQJAF PI=>A@IB DGFQ@CHAJjDQ@IKnLN>AR ~Θ
KnL�DGR�LmDQRT>jRTPC@ILA��=%FQ@CBQR�F Z\[x=%��JARΘk
R�L� (KnBQ=%LmD(,
Θk = a0 + a1N1 + a2N2 + · · ·+ akNk< RUHAJjD�R�B Ds=vPIKnFEB > #��2£v=%PIJAR�F�PIR�Bai
>AR �'= � KnL"4ODGFEKnJj£nR�FzJjL Θk
PIR APCJAB�FQ��=%PI@CBEDGR �KnBEBQ@CHAPIRv8 1 K%JAF-Z�R�PI=6&%K%LqJ9DG@IPC@IBER PIR Z�FQ@;DGr�FQR ��JA@AZ�KnLABQ@CBEDQR94UMq@ILj@IMq@IBQR�F
Q = E[(Θk−Θk)
2],
Q = E[(Θk − Θk)2]
= E[(a0 + a1N1 + a2N2 + · · ·+ akNk − Θk)2]
= E[
k∑
j=0
ajNj −Θk)2]
� KnJ9Dz>0#W=%H(KnFE>�LjKnJAB�>A��DQR�FEM0@CLAKnLABa0,:%:
δEδa0
= 0
⇔ E[2(a0 + a1N1 + a2N2 + · · ·+ akNk −Θk)] = 0
⇔ a0 + E[∑k
j=1 ajNj]− E[Θk] = 0
⇔ a0 +∑k
j=1 ajE[Nj]− E[Θk] = 0
⇔ a0 = E[Θk]−∑k
j=1 ajE[Nj]
# L� (R�J9D >AK%LAZ�FER�Mq APu=vZ�R�Fa0
>x=vLAB0P$#�����Jx=vDG@CKnLg>ARb>A�� x=%F D x=vF0P$#�R- j jFQR�BEBQ@CKnLDGFQK%Jj£n��RSR2DzK%L�K%HjDG@CR�LmD(,
Q = E[(Θk − Θk)2]
= E[(a0 + E[
k∑
j=1
ajNj]−Θk)2]
= E[(E[Θk]−
k∑
j=1
ajE[Nj] + E[
k∑
j=1
ajNj]−Θk)2]
= E[(
k∑
j=1
aj(Nj − E[Nj])− (Θk − E[Θk]))2]
b=v@ILmDGR�Lx=%LmD�LAKnJAB�>A��FQ@C£%KnLAB� x=%F FQ=% A (KnFED�4ai
,
δEδai
= 0
⇔ E[2(Ni − E[Ni])(∑k
j=1 aj(Nj − E[Nj])− (Θk − E[Θk])] = 0
⇔∑k
j=1 ajcov(Nj,Ni) = cov(Θk,Ni)
⇔ cov(Θk,Nk) = cov(Θk,Ni)
# LN>AKn@;D >AK%LAZ FQ��BQKnJA>jFQR PIRYB � BEDQr�MqR > #�����Jx=vDG@CKnLNBQJj@C£v=%LmD� �K%JAF >A�2DGR�FEM0@CLAR�FyPIR�Bai
AFEKn AFER�B54 Z\[A=%��JARΘk
,
(3.1)
∑k
j=1 ajcov(Nj,N1) = cov(Θk,N1)∑k
j=1 ajcov(Nj,N2) = cov(Θk,N2)888 888∑k
j=1 ajcov(Nj,Nk) = cov(Θk,Nk)
:�7
� R�Z�@ FQR2£/@IR�LmDz>AKnLAZ 4 FQ��BQKnJj>AFQRUPIR�B�����Jx=vDQ@IKnLAB�LAK%FQM0=%PIR�B',
E[Θk] = E[Θk]
cov(Θk,Ni) = cov(Θk,Ni) ∀i = 1, . . . ,k
��K%DGKnLjB �/JjR�P%#¢KnL �R�JjD5R�LjZ�KnFER >j��£nR�PIKn j ���PCR�B�DGR�FQMqR�Bcov(Nj,Ni)
R2Dcov(Θk,Ni)
&Z�R�PI= �KnJjF�DGKnJjD @��?:�&C8;8C8�& �(8n<5R �'=v@CD���JAR
N |~ΘBEJA@C£%R�JjLARzPCKn@jMtJAPCDQ@ILAKnMq@u=vPIR-LAKnJAB- (R�F w
M0R2D�>ARU>A@IFERT��JARUP$#�KnL�=?P%#¢���n=%PI@;DG��,
cov(Nj,Ni) = E[cov(Nj,Ni|~Θ)] + cov(E[Nj|~Θ],E[Ni|~Θ])
= E[NΘi(δij − Θj)] + cov(NΘj,NΘi)
= NE[Θi(δij − Θj)] + N2cov(Θj,Θi)
XYRUMqR�MqR ��JAR�,
cov(Θk,Ni) = E[cov(Θk,Ni|~Θ)] + cov(E[Θk|~Θ],E[Ni|~Θ])
= 0 + cov(Θk,NΘi)
= Ncov(Θk,Θi)
� L�FQR�Mq API= � =%LmD�PCR�B�FQ��BQJAP;D\=vDQB�K%HjDGR�L/JjBY>x=%LjB�PCR�B�B � B DGr�MqR �1ij8;: �2&xR2D�R�LbBE@IMq API@ w�¦=%LmD� x=%F � &AK%L�KnH9DG@IR�LmD(,
(3.2)
∑k
j=1 aj{E[Θi(δij − Θj)] + Ncov(Θj,Θ1)} = cov(Θk,Θ1)∑k
j=1 aj{E[Θi(δij − Θj)] + Ncov(Θj,Θ2)} = cov(Θk,Θ2)888 888∑k
j=1 aj{E[Θi(δij − Θj)] + Ncov(Θj,Θk)} = cov(Θk,Θk)
# L KnHjDQ@IR�L�DU>jKnLAZ?PIRtB � BEDQr�MqR �'ij8W7*�z>AR �b����Jx=vDG@CKnLAB 4 ��@ILAZ�KnLAL�JAR�B�8 # L�BQ=%@CD�/J0#�@CPAR2 9@IB DGR JALjR BQKnJAP;DG@CKnLOJALA@C��JAR BQ@API=YM0=vDGFE@IZ�Ry>AR Z�K�£v=%FE@u=%LAZ�R >AR ~N
R�BED-FE���nJAPC@Ir�FQR%8b=%@IBt@IP FQR�B DGR 4�DQFQKnJ9£nR�F?PI=bBQK%PIJjDQ@IKnL�8 1 KnJAF?Z�R�Pu=bLjKnJAB =%PCPIKnLABTLAKnJjBO@ILmDG��FQR�BQBQR�F 4JAL{B � B DGr�M0Rq APIJjB BQ@IMq APCR%8 Xs=vLABUPIR?Z~=vBS x=vFEDG@CZ�JAPC@IR�FsK*) ~Θ
=NJALARq>A@CBEDGFE@IHAJ9DG@IK%L >ARXY@IFE@IZ\[APCR�D(&©P%#¢R�BQ (��FQ=%LAZ�R�>AR
Θk
>AKnLjLA� ~NR�BEDqPI@CLA�~=v@IFQR8&©Z8#�R�BED0>AKnLAZ����n=%PIR�M0R�L�D PI=
FQR��%FQR�BEBQ@CKnL�>jRΘk
BEJAF ~N8
:~i
��������� ��� ����������� �����Θk
�����Nk# L �K%BQR?PIR? jFQKnHAPCr�MqROBQ@CM0 jPI@ �A�tBQJj@C£v=%LmD(&�Z�KnFQFER�BE �KnLj>x=%LmD 4�PI= BQKnPCJjDG@CKnL >x=%LjB
PIR Z~=vBzXz@IFQ@CZ\[APIR2D(,
Θk = bk + ckNK� L� AFQK/Z���>x=%LmD�Z�KnMqMqRs>x=%LAB�PCR Z~=%B��n��LA��FQ=%P�KnL�KnHjDQ@IR�LmD PCR�B�>AR�J! N£v=%PIR�JAFQB BEJA@;w£%=vL�DQR�B� �K%JAF
bk
R2Dck
,bk = E[Θk](1− ckN)
ckvar(Nk) = cov(Θk,Nk)
⇔ ck = cov(Θk,Nk)var(Nk)
⇔ ck = var(Θk)Nvar(Θk)+E[Θk(1−Θk)]
# L� �K%BQR�,ek =
E[Θk(1−Θk)]
var(Θk)# L�=0=vPIKnFEB',
ck =1
N + ek
bk =ek
N + ek
E[Θk]
⇒ Θk =Nk + ekE[Θk]
N + ek
=N
N + ek
NK
N+
ek
N + ek
E[Θk]
# L KnH9DG@IR�LmD >AKnLjZsJALARsR- 9 AFQR�BQBQ@CKnL�=%Lx=%P � DG@I��JARz>AR Θk
&jZ�RY�/Jj@ BQ@C�nLA@��xRY��JARsP$#�K%L= JjLA@I��JAR�M0R�L�D�>A�2DGR�FQMq@ILA�
Θk
>x=vLAB�PIR Z~=vB� A=%FEDQ@IZ�JjPI@IR�F K*)�@CP L # � =?��J #�JjL�BER�JAP NkZ�KnLAL�J �'@IR8,Θk = bk + ckNk
� 8�9@(P%#¢KnL R�BQBQ= � RU>ARsBQRYFG=%MqR�LAR�F =%JNZ�=%B AFE��Z���>AR�L�D�K�) Θk
BQJA@;D JALjRs>A@IB DGFE@IHAJjDQ@IKnL>ARTXY@IFE@IZ\[APCR�D�KnL�KnHjDQ@IR�L�D',
ek = α x=%F JALAR->AR�B5 AFQKn jFQ@I�2DG��B >AR�Pu= PCKn@v>jRyXz@IFQ@CZ\[APIR2D���JAR-P%#¢KnLU= £/JT=%JUZ�KnJAFEB5>AJU>AR�J! 9@Cr�MqRZ\[x=% A@;DGFQRv8 � @ILA=%PIR�M0R�L�D KnLNDGFQK%Jj£nR8,
:��
Θk =Nk + αE[Θk]
N + α
=N
N + α
Nk
N+
α
N + αE[Θk]
��R�M0=%FQ��JAR8,9R�LN�n��Lj��FG=vPek
>A�� (R�LA>�>AR �¦8 �.#¢R�BED �EJjBEDGR�M0R�LmD PIR9�'=%@CDy��JAR�&9PIRzFG=% 9w �KnF DqR2 9 AFQ@CM0�� x=vF
ek
BEKn@CDqZ�K%LABEDG=%LmD(&-��JA@�FQR�Lj>�Pu= >A@CBEDQFQ@IHjJjDG@CKnL >AR�XY@IFE@IZ\[APCR�D BQ@ x=%FEDQ@IZ�JAPI@Cr�FQRv8 1 =vF�R- 9R�Mq APIRUBQ@ P$#�K%L�Z�R�BEBQRt>jRTBEJA A (KnBQR�Fz��JAR
Θk
BQJA@;D�JALART>A@IB DGFQ@CHAJ9wDG@IK%L�>jRSXz@IFQ@CZ\[APIR2D KnL�= ,
Θ1 ≡ 1/2 ⇒ e1 = +∞ et ck = 0
��������� ��� ����������� �����Θk
�����N1,N2, . . . ,Nk��R�£%R�LAKnLjB 4 LAKnJj£%R~=%J =vJ�Z~=vB0�n��LA��FQ=%P 8 # L =�£/J���JAR� (KnJAFqJALAR�FQR��%FQR�BEBQ@CKnL
PI@ILj�~=%@CFQR =~£%R�Z�JALAR�BQR�JAPNk
>AK%LALA��KnLq=%FQFE@C£%R94UK%HjDGR�LA@IF�JjLAR�BQKnPCJjDG@CKnL�8 �=%@ILmDGR�Lx=%LmD(&�'=%@IBEKnLAB Pu=?FER��nFER�BQBE@IKnL�PI@CLA�~=v@IFQRzBQJAF
N1,N2, . . . ,Nk
,
Θk = ak0 +
k∑
j=1
akjNj
KnLb=?PCR B � BEDQr�MqRSBEJA@C£v=%LmD�4 FQ��BEKnJA>AFER�,
(3.3)
∑kj=1 akjcov(Nj,N1) = cov(Θk,N1)
∑k
j=1 akjcov(Nj,N2) = cov(Θk,N2)888 888∑k
j=1 akjcov(Nj,Nk) = cov(Θk,Nk)<5= ��JAR�B DG@IK%L0��JAR�P%#¢KnL BQR� (KnBQR�R�B D >jR�BQ=~£nKn@CF�BQ@¦Pu=sBEKnPIJjDQ@IKnL?>jR�Z�R�DEDGR�FQR��%FQR�BEBQ@CKnLR�BEDz@I>jR�LmDG@C�/JjR 4qPI=O AFQ��Z���>jR�LmDGRv8 � RU��JA@�£nK%JA>AFG=v@CD�>A@IFER ��JAR�,
akj = 0 ∀j = 1, · · · ,l j 6= kakk = ck
ak0 = bk� BED wlZ�RzJALjRzBEKnPIJjDQ@IKnL >AJ0B � BEDQr�MqR��'ij8Wi*���3�.#¢R�B D Z�R���JAR�P%#¢KnLq£v= DGR�L�DQR�Fy>ARz jFQKnJ9w£nR�F�81 KnJAFYPI=
keme ���/JA=vDG@CKnLd>AJdB � BEDGr�M0R8& x=vBY>ARO jFQKnHAPCr�MqR�&¦P%#¢���n=%PI@;DG� R�B Ds£n��FQ@��f��R%8� L�R���R�D',
:�V
akkcov(Nk,Nk) = cov(Θk,Nk)
⇔ ckvar(Nk) = cov(Θk,Nk)
⇔ ck = cov(Θk,Nk)var(Nk)
� RU��JA@ R�B D�Pu=?>A���ALA@CDQ@IKnLN>ARck
8+ K � K%LAB�Z�RT�/JjRTP$#�K%L�K%HjDG@CR�LmD� �K%JAF i 6= k
,
akkcov(Nk,Ni) = cov(Θk,Ni)
⇔ akk = Ncov(Θk ,Θi)N2cov(Θk ,Θi)−NE[ΘkΘi]
⇔ akk = 1N+fik
�9@ PIR�Bakj
(KnBE��B�BQKnLmD �EJAB DGR�B�KnL�>AKn@;D�=~£nKn@CF',
fik =−E[Θk(1− Θk)]
cov(Θk,Θi)= ek
b=vPI[AR�JAFQR�JABQR�M0R�L�DTZ�R?L #�R�BEDT x=%B PCR?Z~=%B�8 � L R2� R2D(&�BEJA A (KnBQK%LABU�/JjRq (KnJAF JjL�@Θi
BEKn@CD�>A�2DGR�FQMq@ILA�8&AKnL�=0=vPIKnFEB JAL�Z�KnLmDGFERUR2 9R�Mq APIR8,
Θi
>A�2DGR�FEM0@CLA�⇒ fik = +∞, akk = 01 =%FzZ�KnLmDGFER�&(BQ@5P%#¢KnL�BQRt (R�LjZ\[AR 4qLAKnJj£%R~=%JdBEJAF�PIRUZ~=%BYXY@IFE@IZ\[APCR�D�KnLb=0 x=vFzJALjR
>AR�B� AFQK% AFQ@C��DG��B�R�LAKnLjZ���RT>x=%LjB�PCR >AR�J j@Cr�MqR Z\[x=% A@;DGFER�,
−E[Θk(1− Θk)]
cov(Θk,Θi)= α =
E[Θk(1− Θk)]
var(Θk)� Rq��JA@©Z�KnL6�AFQMqR?PIR �'=%@;Ds��JAR Z�R2DQDGR? jFQKn AFE@I�2DG�?L #¢R�B DS A=%Bs£n��FQ@ �A��R PIKnFEBQ��JAR?P$#�KnL LjRBQJA A (KnBERS x=vB���JAR ~Θ
BQJA@;D�JALARU>A@IB DGFE@IHAJjDQ@IKnLN>jRSXz@IFQ@CZ\[APIR2D~8
������� � �� ��� ��� � �
� J Z�KnJAFQBT>ARqZ�RqZ\[x=% A@;DGFERqK%L�=�>A��DQR�FQMq@ILj�qJjLAR0BEKnPIJ9DG@IK%L >ARΘk
(KnJAFUPIR Z~=vBBQ@IMq APCRt> #¢JALAROFE���nFER�BEBQ@IK%L�BEJAF
Nk
& jJA@IBsK%L = BQJA j �KnBE�?��JAR?Pu= BEKnPIJ9DG@IK%Lb (KnJAFsJALjRFQR��%FQR�BEBQ@CKnL�>jR
Θk
BQJjFN1,N2, . . . ,Nk
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BEJA@CD�JjLARSPCKn@ >jRSXz@IFQ@CZ\[APIR2D(&API=OBQKnPCJjDG@CKnLN>AJ�Z~=%B�BE@IMq APIRsR�BED�£n��FQ@ �f��R �KnJjF�JjLARUFQR��%FQR�BEBQ@CKnL�BEJAF
N1,N2, . . . ,Nk
8:~�
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>AR DGFEKnJj£%R�F�JALjR BQK%PIJjDQ@IKnLt=%JO AFEKnHAPIr�M0R >ARyPu=zFER��nFER�BEBQ@IK%L?>ARΘk
BEJAFN1,N2, . . . ,Nk
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BEJA@CD�JjLAR >A@CBEDGFE@IHAJ9DG@IK%L>ARTXY@IFE@IZ\[APCR�D~8
<5=UFQR�Z\[jR�FQZ\[jR > #�JjLARYBEKnPIJjDQ@IKnL BQ=%LAB P$#�[ � (K%DG[Ar�BQRYXY@IFE@IZ\[APCR�D �K%JAFQFQ=%@CD �'=%@CFQR�P%#¢KnH9w�ER�D�> #¢JALARU��DQJA>ART>A@ ����FER�LmDGRU��JARUZ�R�PIPIR ��JA@ R�BED��'=%@CDQRs (KnJAF PIR Z\[A=% A@CDQFQRUBQJj@C£v=%LmD~8
:�]
� � �����( ���� �
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Nj
>AR Z\[x=%��JARNj :=
∑li=1 Nij
R�LJjDG@CPI@IBQ=%LmD�PIR�B�>jKnLALA��R�B��/JjRtP%#¢KnL�=�4q>A@CBQ (KnBQ@;DG@CKnL�8A< RU AFE��BQR�LmDsMqK9>Ar�PIRs£/@IBERS>jKnLAZ 4KnHjDGR�LA@IF JjLAR�DGR�PIPIR©R�BEDQ@IM0=vDG@CKnL�8�< R�B5[ � (K%DQ[Ar�BER�B" �KnJjF Z�R�MqK9>Ar�PIR�BEKnLmD"PIR�B BQJA@;£%=vL�DQR�B�, 1 KnJjF�Z\[A=%��JARt=vLALA��RT> #¢K9Z�Z�JAFER�LAZ�R7��&
:n8���D\=%LmDT>AKnLALj��BNj
R�D ~Θj = Θ1j, . . . Θlj
& KnL =���JAR(N1j , . . . ,Nlj) ∼
multinomiale(Nj ,~Θj)8
798Nj
R2D ~Θj
BQKnLmD�@ILA>j�� (R�LA>x=vL�DQB�8 1 KnJjF�P$#�R�LABQR�MOHAPCR >AR�Bz=%LALA��R�B�> #¢K9Z�Z�JAFER�LAZ�R�&
:n8�PIR�Bs£nR�Z2DGR�JAFQBS=%PI��=vDGK%@IFQR�B ~Θ1, . . . ,~Θl
BQKnLmDT@ 8¢@ 8¢>dR2DSBEJA@C£%R�LmDTZ\[x=%Z�JAL{JALjR>A@IB DGFE@IHAJjDQ@IKnL >ARTXY@IFE@IZ\[APCR�D�>ARU x=%FQ=%Mqr�DGFER�B
(α1, . . . ,αl)8
798�< R�BS£v=%FE@u=%HjPIR�BS=%PC�~=vDQKn@IFER�BNj
BQKnLmDt@ILA>A�� �R�LA>x=%LmDQR�B?R�Dt �K%BQBQr�>AR�LmD JALjR>A@IB DGFE@IHAJjDQ@IKnL >jR 1 Kn@IBEBQKnLg>AR� x=%FQ=%Mqr�DGFER
Vjλ&�K �
Vj
R�BED0JAL £nKnPCJAM0RZ�KnLjL/J 8 # L�= =%PIK%FQB ��JAR
E(Nj) = var(Nj) = Vjλ8
# L�DGFQ=~£%=v@IPIPCR�>j��BQK%FQM0=%@IB (KnJAF JALAR Z�K%PIKnLALjRj>A�2DGR�FQMq@ILA��RdR�D�KnL LAKvDGRd x=vF
k := l−j +1PCR LAKnMOHAFER > #¢��PI��M0R�LmDGB->AR�Z�R2DQDGR�Z�KnPIKnLjLAR%8��9=%Z\[x=%LmD Z�R�Z�@%&%KnL0L0#���Z�FQ@CFG=
APIJAB P%#¢@ILj>A@IZ�R �- x=%F-PI=UBEJA@CDQR �Nj
BER�FG=T>AKnLAZ�LAK%DQ�� x=%FNR�DyPIR�B->AK%LALA��R�B
Nij
BQR�FEKnLmDLAK%DG��R�B� x=%F
Ni
�28��KnJAB =~£nK%LAB Z\[AKn@CBQ@�>ARtZ\[AR�FEZ\[AR�FUJAL R�BEDG@CM =3DGR�JjFsPI@CLA�~=%@CFQRUR�L PCR�Bs>AK%LALA��R�B ��JAR
LAKnJAB� (KnBQBE��>AKnLjB'&(Z�#¢R�BED54 >A@IFER JAL�R�B DG@IM0=vDQR�JAF >ARUPI=��fK%FQMqRsBQJA@;£v=%LmDGR�,
N = a0 +k
∑
i=1
aiNi
:~�
R�D�KnL�@IMq �KnBERs��JARUZ�R2DYR�BEDG@CM =3DGR�JjF Kn jDQ@IMq@IBQRYPIR Z�FE@CDQr�FQR BEJA@C£v=%LmD',Mq@ILA@CM0@CBQR�F
E[(N −N)2]
<5RqHAJjDTR�BEDS>AKnLAZq>AR DGFQK%Jj£nR�FSP$#�R- j jFQR�BEBQ@CKnL >jR�Bai
R�L �fKnLAZ2DG@IK%L�>jR�BTM0KnMqR�LmDQBT>jRPu=�>A@CBEDGFE@IHAJ9DG@IK%L >AR
NR�D�R�L �fKnLjZ�DG@CKnL >AR�B� x=vFG=%Mqr�DQFQR�B
αi
�'��JA@s>A�2�xLA@CBQBQR�LmDbPI=>A@IB DGFQ@CHAJjDQ@IKnL >AJ � £%R�Z2DGR�JjFN>ARbFE@IBQ��JAR�� ~Θ
� �K%JAF ��JARdPI= Z�KnLA>A@;DG@IK%L�> #¢Kn jDQ@IM0=%PI@;DG�@IMq �KnBE��RsBEKn@CD FQR�M0 APC@IRv8 � LN> #�=%JjDQFQRsDGR�FQMqR�B�&ALAKnJAB �'=%@IBEKnLAB JALjR FQ���%FQR�BEBQ@CKnLNPI@CLA�~=%@CFQR>AR
NBQJAF
N1, . . . ,Nk
8< R AFEKnHAPIr�M0R R�BED�>AKnLjZTPCR BQJA@;£v=%LmD(,
Z\[AR�FEZ\[AR�FN = a0 +
k∑
i=1
aiNi
D�8��¦8
E[(N −N)2]BQKn@;D�M0@CLA@IM0=%PIR
�� � � � ����� � �" � " � � �" ������� ����� �� ��KnJABz=�£%KnLAB�£/Jb>x=%LjBYPIR Z\[A=% A@CDQFQRT AFQ��Z���>AR�LmDs��JARSZ�R�Z�@5FQR2£9@CR�LmD 4 FQ��BQKnJj>AFQRTPIR�B
����Jx=vDQ@IKnLAB�LAK%FQM0=%PIR�B BQJA@;£v=%LmDGR�B�,
E(N) = E(N)�'�A8;: �
cov(N ,Ni) = cov(N,Ni) �KnJjF
i = 1, . . . ,k�f�A8W7*�
� L�>A��£%R�PIK% A x=%LmD�P%#¢����Jx=vDQ@IKnL �f�A8C:/� KnL�KnHjDG@CR�LmD(,
E(N) = E(N)
⇔ E(a0 +k
∑
i=1
aiNi) = E(N)
⇔ a0 = E(N)−
k∑
i=1
aiE(Ni)
⇔ a0 = E(N)−
k∑
i=1
aiE(E(Ni|~Θ))
⇔ a0 = E(N)−
k∑
i=1
aiE(NΘi)
:~p
⇔ a0 = E(N)−
k∑
i=1
aiE(N)E(Θi)
⇔ a0 = E(N)(1−
k∑
i=1
aiE(Θi))
⇔ a0 = E(N)(1−k
∑
i=1
ai
αi
α)
XY�2£nR�PIKn A (KnLAB�M0=%@ILmDQR�Lx=%LmD�P$#����/JA=vDG@CKnL �'�A8W7*�2,
cov(N ,Ni) = cov(N,Ni) �KnJjF
i = 1, . . . ,k
⇔ cov(a0 +k
∑
j=1
ajNj,Ni) = cov(N,Ni) �KnJjF
i = 1, . . . ,k
⇔k
∑
j=1
ajcov(Nj,Ni) = cov(N,Ni) �K%JAF
i = 1, . . . ,k
1 KnJAF�PCRsDGR�FEM0RU>ARU�m=vJAZ\[AR�&xK%Lb=6,
cov(Nj,Ni) = cov(E(Nj|~Θ),E(Ni|~Θ)) + E[cov(Nj,Ni|~Θ)]
= E(N2ΘjΘi)− E(NΘj)E(NΘi) + E[N(−ΘjΘi + δijΘi)]
= E(N2)E(ΘjΘi)− E(N)2E(Θj)E(Θi)− E(N)[E(ΘjΘi)− δijE(Θi)]
=
{
E(N2)αiαj
α(α+1)− E(N)2 αiαj
α2 − E(N)αiαj
α(α+1)
�K%JAFi 6= j
E(N2)αi(αi+1)α(α+1)
− E(N)2 α2i
α2 − E(N)(αi(αi+1)α(α+1)
− αi
α)
�K%JAFi = j
1 KnJAF�PCRsDGR�FEM0RU>ARU>AFEKn@CDQR�&AK%L�=!,
cov(N,Ni) = cov(E(N |~Θ),E(Ni|~Θ)) + E[cov(N,Ni|~Θ)]
= cov(N,NΘi)
= E(N2Θi)− E(N)E(NΘi)
= E(N2)E(Θi)− E(N)2E(Θi)
= E(Θi)var(N)
=αi
αvar(N)
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7v}
HAPIK/Z?: 8888 8 8 HjPIK/ZU7
. . .
. . .HAPIK/ZT7 8 8 8
888 HjPIK/Z?:
a1
a2888ak
=
α1
αvar(N)
α2
αvar(N)888
αk
αvar(N)
K ��PCR�B ��PI��MqR�LmDQB�>jJ�HAPCK9Zt:SBEKnLmDz>jJ�D � (R�,
E(N2)αi(αi + 1)
α(α + 1)− E(N)2 α2
i
α2− E(N)(
αi(αi + 1)
α(α + 1)−
αi
α)
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E(N2)αiαj
α(α + 1)− E(N)2 αiαj
α2− E(N)
αiαj
α(α + 1)=�£%R�ZSPIR�B @CLA>A@CZ�R�B�@�R2D ��FQR�BQ (R�Z2DG@C£%R�MqR�LmDY=%>x=% jDQ��Bz=%J! �@CLA>A@CZ�R�B >jRTPI=?PI@I�%LARsR�D�>ARUPI=Z�KnPCKnLALART=%J! 9��JAR�PIPIR�B BQRUDGFEKnJj£nRUPCRsDGR�FEM0Rv8
� DTP%#¢����Jx=vDQ@IKnL �f�A8C:/�YLAKnJABT>AKnLALjRqPI=NZ�KnLA>j@CDG@CKnL �R�FQMqR�DED\=%LmDU>ARq>A��>jJA@IFERa0
4 x=%FEDQ@IF >AR�B
ai
,
a0 = E(N)(1−k
∑
i=1
ai
αi
α)
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4�DGFEKnJj£%R�FOJAL{R�BEDQM =3DGR�JjFN
>jR Pu= �fKnFQMqRN = a0 + a1N1
8$Xs=%LjBOZ�R Z�=%B'&PIRSB � BEDQr�MqRO> #¢����Jx=vDQ@IKnLABYR�DsPI= Z�K%LA>A@CDQ@IKnL� (KnJAF a0
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a1(E(N2)α1(α1+1)α(α+1)
− E(N)2 α21
α2 − E(N)(α1(α1+1)α(α+1)
− α1
α) = α1
αvar(N)
a0 = E(N)(1− a1α1
α)
7/:
Z�#¢R�B D34?>A@CFQR�,
a1 =var(N)
E(N2)α1+1α+1
− E(N)2 α1
α+ E(N)α−α1
α+1
a0 = E(N)(1− a1α1
α)
Xs=%LAB�Z�RUZ�=%B'&xPI=OBQKnPCJjDG@CKnLNR�B DY>AK%LAZU x=%F �'=v@CDGR�M0R�LmD >A��DQR�FEM0@CLA��Rv8798�< R >jR�J! 9@Ir�M0R-Z�=%B5Z�KnLABE@I>A��FQ� R�B D$Z�R�PCJA@m>AR-Pu= AFER�Mq@Ir�FER�Z�KnPCKnLALAR8&~Z�#¢R�B D 4�>A@CFQR�PIR
Z~=%B K��NKnLN �K%BQBQr�>AR DGKnJ9DGR�B PCR�B K%HABQR�FE£v=vDG@CKnLAB LjR�Z���BQBQ=%@IFER�B�8 # LNZ\[AR�FQZ\[ART>AKnLjZM =v@ILmDGR�Lx=%LmD"JALSR�BEDQ@IM0=vDGR�JAF >AR Pu= �fKnFQMqR
N = a0+∑l
i=1 aiNi
8 �=%@IB5>x=%LjB$Z�RZ~=%B�&�KnL0BQ=%@CD-��JAR�P%#¢R�B DG@CM =vDQR�JAF©�/JjRzLAK%JAB->AR�£%KnLAB K%HjDGR�LA@IF�R�BED
N =∑l
i=1 Ni
& AJA@CBQ��JAR P$#�K%L Z\[AR�FQZ\[AR 4�R�BEDG@CM0R�FSPI=NBQKnMqM0R >AR�Bt>AK%LALA��R�BtR2Dt��JAR0LAKnJABTPIR�B �K%BQBQ��>AKnLAB�DGK%JjDGR�B�8
� L�DQR�Lx=%LmD Z�KnMq jDGR�>AR�Z�R�B >AR�J! Z�=%B BE ���Z�@u=vJ! &yKnL� �R�JjD (KnBQR�F �f �K%JAF JALjRZ�KnPCKnLALAR � �/JjR�PIZ�KnLA��JART �K%BQB �>A=%LmD � ��PI��M0R�LmDGB�� JALjRTBEKnPIJ9DG@IK%LN>ARUPu= �fKnFEM0R8,
ai =var(N)
E(N2)∑
αm+1α+1
− E(N)2∑
αm
α+ E(N)α−
∑αm
α+1
a0 = E(N)(1−k
∑
i=1
ai
αi
α) �K%JAF
m£v=%FQ@I=%LmD >AR : 4 � R�D� (KnJAF
i = 1, . . . ,k
# L?FER�DQFQKnJj£%R HA@IR�L =%PIK%FQB©PI=YBQK%PIJjDQ@IKnLAB$>AR Z\[x=vZ�JAL >AR�B�>AR�J! OZ~=%B�BE ���Z�@I=%J! tPIKnFEBQ��J #¢KnLFQR�Mq API=%Z�R �� A=%FS:tR�DsP5FER�BQ (R�Z2DG@;£nR�MqR�LmD�8 � Rt��JA@$R�BEDY��DQKnLALx=%LmDY>AROFQR�M =vFQ��JAR�F�R�B D�/JjRdP%#¢KnL � KnHjDQ@IR�L�D � �'KnJ ��JARdP%#¢KnLgBQJA j �KnBERdBQ@CM0 jPIR�MqR�LmDq (KnJAFqPIR�M0KnMqR�LmD�� JALjRBQKnPCJjDG@CKnL�=~£nR�ZTPIR�B
ai
DQKnJAB����m=vJ! ��
�� � � � ���� ������ ������� �" � ��� �!�������� � �" ��� ��� " ������ ������ �
klP/LAKnJAB �'=%JjD$HA@IR�LOR�LmDQR�LA>AJ?R�LAZ�K%FQR >A��M0K%L�DQFQR�F©��JAR PI=zBEKnPIJjDQ@IKnLT��JAR LAK%JAB�=~£nK%LAB��LAKnLjZ���RqR�B DTZ�KnFEFQR�Z2DGRv8 1 KnJAF Z�R�DQDQR0>j��MqKnLABEDQFG=vDQ@IKnL0& LAKnJjBS=~£%KnLABUH�R�BQKn@CL >AR >AR�J! FQ��BEJAPCDG=vDGB �/JjRsLAKnJAB �fKnJAFQLj@IBQBER�LmD PCR�B jFQKn (KnBQ@;DG@CKnLAB�:sR�D 798/£%Kn@IZ�@ Z�R�B >AR�J! F��QJjPCD\=3DGB',
:n8��9@(N1, . . . ,Nl|~Θ) ∼ multinomiale(Θ1, . . . ,Θl,N)
&�=%PCKnFQB(∑k
i=1 Ni,∑l
i=k+1 Ni) ∼
multinomiale(∑k
i=1 Θi,∑l
i=k+1 Θi,N)8�� R-FQ��BQJAP;D\=vD5R�BED �fKnJAFELA@n x=vF"PI= AFQK3w
�K%BQ@CDQ@IKnL :%8
7%7
798��9@(Θ1, . . . ,Θl) ∼ Dirichlet(α1, . . . ,αl)
&�=%PCKnFQB(∑k
i=1 Θi,∑l
i=k+1 Θi) ∼
Dirichlet(∑k
i=1 αi,∑l
i=k+1 αi)8*� R FE��BQJjPCD\=3D�R�B D �fKnJAFELA@/ x=%F�PI=z AFEKn �K%BQ@CDQ@IKnL
798klP�LAK%JAB �'=%JjDT> #�=%H�K%FQ>�>A��MqKnLmDGFER�FU��JAR0PCR�B
ai
BQKnLmDSHA@IR�L DQKnJABT���m=vJ! & Z�KnMqM0RZ�RT�/JjRTP$#�K%L�=~£v=%@;D BQJA j �KnBE�%8 1 K%JAF�Z�R�PI=6&AKnL�=?H�R�BQKn@CL�>AJ�Z�KnFQK%PIPu=v@IFQRzBQJA@;£v=%LmD(,¤t�¦���(a1af�x&f��, � ��� � � � ��� � � ��� � �
N =∑l
i=1 Ni
����� � ��� � N1, . . . ,Nl� � ��� � � ��� ��� � � ���
� ��� ��� � � � � ��� ��� � ��� � � � � (α1, . . . ,αl)� � � ������� � � ��� � � ��� � � Mk =
∑k
i=1 Ni
� ��� ���� � N1, . . . ,Nk
� � ��� � � ��� ��� � � ��� � ��� ��� � � � � ��� ��� � ��� � � � � (α1, . . . ,αk) ��S�������XYKnLALA�
N = n,
��D\=%LmD >jKnLALA�Mk
R2D ~Θ&/=%PCKnFQB-KnL =S��JAR
(N1, . . . ,Nk)BEJA@CD JALARz>A@IB DGFE@IHAJjDQ@IKnL
MOJAP;DG@ILjKnM0@I=%PIR >ARb x=vFG=%Mqr�DQFQR�B(Mk,
Θ1
Θk, . . . ,Θk
Θk)& K �
Θk =∑k
j=1 Θj
8 � LR���R�D'&xKnLb=!,
Pr[N1 = x1, . . . ,Nk = xk|Θk = m,~Θ]
=Pr[N1 = x1, . . . ,Nk = xk|~Θ]
Pr[Θk = m|~Θ]
=~£%R�Zx1 + . . . + xK = m
=
n!x1!···xk!(n−m)!
Θx1
1 · · ·Θxk
k (1− Θk)n−m
n!m!(n−m)!
Θmk (1− Θk)n−m
=m!
x1! · · ·xk!(Θ1
Θk
)x1 · · · (Θk
Θk
)xk
�.#�R�BEDzHA@IR�L�Z�RT�/JjRTP$#�K%L�£%KnJAPI=%@CD MqKnLmDGFER�F�8 ��D\=%LmD >AK%LALA�
Θk = y&�KnLd= �/JjR
(Θ1
Θk· · · Θk
Θk)BEJA@CD JjLAROPIK%@$>AROXY@IFE@IZ\[APCR�Ds>jR
x=%FQ=%M0r2DGFER�B(α1, . . . ,αk)
8 � LUR���R�D'&�BQKn@CR�LmDx1, . . . ,xk
DQR�PCB �/JjRx1+. . .+xk =
1&¦=%PCKnFQB KnLb=!,
dP (Θ1
Θk
= x1, . . . ,Θk
Θk
= xk|Θk = y)
=dP (Θ1 = x1y, . . . ,Θk = xky)
dP (Θk = y)
=
Γ(α)yk−1
∏kj=1 Γ(αj )Γ(α−αk)
(x1y)α1−1 . . . (xky)αk−1(1− y)α−αk−1
Γ(α)Γ(αk)Γ(α−αk)
yαk−1(1− y)α−αk−1
=Γ(αk)
∏kj=1 Γ(αj)
xα1−11 · · ·xαk−1
k
� R�DEDGR R- 9 AFQR�BQBQ@CKnLtLjRy>A�� �R�LA>O APCJAB5>ARy&�R2D"Z�#¢R�BED©HA@IR�LSPI= �fKnLAZ�DQ@IKnLT>AR >AR�LjBQ@CDQ�
> #¢JALARTPIKn@ XY@IFE@IZ\[APCR�D >ART x=%FQ=%Mqr�DGFER�B(α1 . . . αk)
87vi
2
# Ls£v= M =%@CLmDGR�LA=%LmD BQJA j �KnBER�F5��JAR-PIR�Baj
BQKnLmD DQKnJAB ���n=%J! ��a1 = . . . = ak = a
�-&R�D?£nK%@IF?��JARNLAKnJjB?KnHjDGR�LAKnLAB >x=vLAB?Z�RNZ�=%B JALARNBQK%PIJjDQ@IKnL� (KnJAF
a��JA@ LARN>j�� (R�LA>
�EJABEDQR�MqR�LmD x=vB >AR ��8 # L (KnJAFEFG=T>AKnLAZzZ�KnLAZ�PIJAFERz��JARYZ�R�DEDGRzBQKnPCJjDG@CKnL0BQ=vDG@CB �'=%@;D-HA@IR�LLAK%DGFER0 jFQKnHAPCr�MqR�&�R2DTZ�KnMqM0R?LjK%DGFER0 AFEKnHAPCr�MqR? �KnBEBQr�>AR0JjLAR0BEKnPIJ9DG@IK%L�JjLA@I��JAR8& KnL �KnJjFQFG= =%PCKnFQB Z�KnLAZ�PIJAFERT��JARULAKvDGFQRUBEKnPIJjDQ@IKnLNR�BED�Pu=OH(KnLALjR%8
# L� x=%F Dz>jRTP$#�����Jx=vDG@CKnL �'�j8 7*�-,
k∑
j=1
ajcov(Nj,Ni) = cov(N,Ni) (KnJAF
i = 1, . . . ,k
⇔ ak
∑
j=1
cov(Nj,Ni) = cov(N,Ni) (KnJAF
i = 1, . . . ,k
⇔ a · cov(k
∑
j=1
Nj,Ni) = cov(N,Ni) (KnJAF
i = 1, . . . ,k
⇔ a · cov(Mk,Ni) = cov(N,Ni) �K%JAF
i = 1, . . . ,k
⇔ a =cov(N,Ni)
cov(Mk,Ni)
K �Mk =
k∑
j=1
Nj
# L�=~£v=%@CD >j� � 4 Z~=%PCZ�JAPC��,
cov(N,Ni) =αi
αvar(N)
R�D�>AKnLAZ8&x x=%F�PIR Z�KnFQK%PIPu=v@IFQRT:*,
cov(Mk,Ni) =αi
α∗
var(Mk)K �
α∗ =
k∑
j=1
αj
# L�= =%PIK%FQB �xLx=vPIR�MqR�LmD',
a =cov(N,Ni)
cov(Mk,Ni)=
αi
αvar(N)
αi
α∗var(Mk)
=α∗
αvar(N)
var(Mk)
R�D �'R�L�JjDQ@IPC@IBG=vL�D PIR�B FE��BEJAPCDG=vDGB�>AJ�>A��HjJjD�>ARUBQR�Z2DG@CKnL �2,
var(Mk) = var(E(Mk|~Θ)) + E(var(Mk|~Θ))
73�
= var(N
k∑
j=1
Θj) + E[N
k∑
j=1
Θj(1−
k∑
j=1
Θj)]
= E(N2(
k∑
j=1
Θj)2)− [E(N
k∑
j=1
Θj)]2 + E(N)E[
k∑
j=1
Θj(1−
k∑
j=1
Θj)]
= E(N2)E((
k∑
j=1
Θj)2)− E(N)2(E(
k∑
j=1
Θj))2 + E(N)E[
k∑
j=1
Θj(1−
k∑
j=1
Θj)]
= E(N2)α∗(α∗ + 1)
α− E(N)2(
α∗
α)2 + E(N)
α∗(α− α∗)
α(α + 1)
>AKnLAZ
a =α∗
αvar(N)
var(Mk)
=α∗
αvar(N)
E(N2)α∗(α∗+1)α
− E(N)2(α∗
α)2 + E(N)α∗(α−α∗)
α(α+1)
21 KnJAF FE��BQJjM0R�F'&j>A=%LAB PCRsZ~=%B > #�JjL DGFQ@I=%LA�nPCRz>jRs>A��£%R�PCKn A (R�MqR�LmD(&APCRYFE��BQJjPCD\=3D R�B D>AKnLAZUPIR BEJA@C£v=%LmD',
Nj = aj0 + aj1Nl+1−j
=~£nR�ZNl+1−j = N1j + . . . + Nl+1−j,j
aj0 = E(N)(1− aj1
l+1−j∑
i=1
αi
α)
aj1 =var(N)
E(N2)∑
αm+1α+1
− E(N)2∑
αm
α+ E(N)α−
∑αm
α+1
�K%JAFm£v=%FE@u=%LmD >AR
14
l + 1− j
7%V
� � �����( ���� �
� � � � � N� � �
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1 KnJAFtZ�R�PI=bLAKnJABO=�£%KnLAB?Z�KnM0MqR�LjZ�� A=%Ft>A��Z�FE@IFQR0PIR�BO>AR�J � AFE@ILAZ�@I x=vPIR�BT>A@CBEDGFE@;wHAJjDG@CKnL�>AKnLmD�LAKnJABz=~£nKnLAB�H(R�BQK%@IL &xZ8#�R�BED.4q>A@CFQR PI= >j@IBEDQFQ@CHAJjDG@CKnLNMOJAPC@CDQLAKnMq@u=%P¦R�D�PI=>A@IB DGFQ@CHAJjDQ@IKnLN>ARTXY@CFQ@CZ\[APIR2D~8
1 JA@CB'& >x=%LjBSPCRODGFQK%@IBQ@Cr�MqROZ\[x=% j@CDGFER�& LAK%JABt=~£%KnLABTM0@CB R�L{ jPu=%Z�RqJjL MqK9>jr�PIR?>jR AFQ��>A@IZ2DG@IK%Lt>AR ~Θ
=~£nR�Z �g>jKnLALA�v8 � JOZ�KnJjFQB©>jR Z\[x=% j@CDGFER�&�LAKnJjB©LAKnJjB©BQK%M0MqR�B5FQR�LA>AJZ�KnMq jDGR �/JjR P%#¢[ � (K%DQ[Ar�BER �/JjR ~Θ
BQJA@;D�JALAR >j@IBEDQFQ@CHAJjDG@CKnLT>AR XY@CFQ@IZ\[jPIR�D©LAKnJAB© AFQK/Z�JAFER>AR�B AFEKn AFQ@C��DQ��B?��JA@ LAKnJjB? �R�FQMqR�DQDQR�LmD > #¢R2 9 APC@IZ�@;DGR�FOJAL M0K/>Ar�PIR >ARN jFQ��£/@CBQ@IK%Lg>jR~Θ8 � LgR2� R2D(&�LjKnJAB =~£nKnLjB?FQR�M0=%FE�/Jj�N�/JjRNBG=%LAB Z�R�DQDQRN[ � (K%DQ[Ar�BER�&�PIR M0K/>Ar�PCR R�BEDH�R�=%JAZ�K%JA b APCJAB >A@ �0Z�PIR 4?Z~=vPIZ�JjPIR�F�8Xs=vLAB�PIR ��Jx=vDQFQ@Cr�MqR Z\[x=% A@;DGFER�&nLAK%JAB-=~£nKnLAB->A�2£nR�PIKn A (��R�D->A��M0K%L�DQFQ� JALqMqK9>Ar�PIR
>AR0 jFQ��>j@IZ�DQ@IKnLjR?>AR �S85< R�BTZ�KnLjZ�PIJjBQ@IK%LABU>AJ Z\[A=% A@CDQFQR AFE��Z���>AR�LmDtBQJAFUP%#¢[ � �K%DQ[Ar�BERXY@IFE@IZ\[APCR�D(&(LAKnJjB KnLmD Z�KnLj>AJA@CD 4 LAKnJAB @CLmDG��FER�BEBQR�FU=vJ Z~=%B MOJ9DGPI@CLAKnMq@u=%P(w�XY@CFQ@IZ\[jPIR�DR2 9Z�PCJABQ@;£nR�MqR�LmD�8 < RtMqK/>Ar�PCR �/JjRtP%#¢KnL�=qDQFQKnJj£%�t (KnBQBEr�>ARS>AKnLAZS>AR�B� AFEKn AFQ@C��DQ��B��/Jj@>A��Z�KnJAPIR�LmD->A@IFER�Z�DQR�MqR�LmD�>AJ #¢H(KnLOZ�K%M0 (KnFEDQR�MqR�LmD(#v>AR PI=z>A@CBEDGFE@IHAJ9DG@IK%Lt>AR Xz@IFQ@CZ\[APIR2D~8
Xs=vLAB�PCR Z�=%>AFER > #¢JALAR ��DQJA>AR�M0Kn@CLABNDG[A��KnFQ@C��JAR >jJ AFEKnHAPIr�M0R8& @IP BER�FG=v@CD�@IL9wDG��FER�BEBG=%LmD0>ARN>A@IBE �K%BQR�F?>0#�JAL DGFE@u=%Lj�nPIR0>ARN>A��£%R�PCKn A (R�MqR�LmDqZ�KnMq �K%BQ� >AR�>AK%LALA��R�BKnHABQR�FE£%��R�B� x=%F JAL�=%BQBEJAFQR�JAF�8 # LN (KnJAFEFG=%@;D =%@CLABQ@�M0R2DQDGFERsR�LN jPu=%Z�RsJAL�M0K/>Ar�PIRYB D\=3wDG@IB DG@C�/JjR?> #�R�BEDQ@IM0=vDG@CKnL >AR�BU x=vFG=%Mqr�DQFQR�B
αi
R�Dλ>AR LAKvDGFQR MqK9>jr�PIRt>AR AFQ��>A@IZ2DG@IK%L
>AR �S85< #��2£v=%PIJx=3DG@IK%L >AR?Z�R�BS x=vFG=%Mqr�DQFQR�B LjKnJABU �R�FQMqR�DEDGFG=v@CD >AR?DQR�BEDQR�FUPu= £v=%PC@I>A@;DG�>ARULAK%DGFERUM0K/>Ar�PCRs>ARU AFE��>A@CZ�DG@CKnL�8
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