26
~ Θ Θ k N k Θ k N 1 ,N 2 ,...,N k

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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

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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

klP�LjKnJAB�R�BED�JjDQ@IPCRs>ARUM0R�LmDG@IK%LALAR�F PCR�B  AFEKn AFE@I��DQ��B�BQJj@C£v=%LmDGR�B',:n8�< R�B©>j@IBEDQFQ@CHAJjDG@CKnLAB M0=%FQ�%@ILx=%PCR�B�>jR�B$£v=%FE@u=%HjPIR�B5=vPI�~=3DGKn@CFQR�B

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)

]

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(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

� � �����( ���� �

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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

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# 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

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Nj ≡R�D

Nij ≡ Ni

� 8

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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

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# 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

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�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

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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

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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

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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)

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Θk

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:~i

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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]

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Θk

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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

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ek

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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

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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

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>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

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������� � �� ��� ��� � �

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 (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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N1,N2, . . . ,Nk

8:~�

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BEJAFN1,N2, . . . ,Nk

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:�]

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Nj

>AR Z\[x=%��JARNj :=

∑li=1 Nij

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: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�&

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(α1, . . . ,αl)8

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Vjλ&�K �

Vj

R�BED0JAL £nKnPCJAM0RZ�KnLjL/J 8 # L�= =%PIK%FQB ��JAR

E(Nj) = var(Nj) = Vjλ8

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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

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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

:~�

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E[(N −N)2]

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NR�D�R�L �fKnLjZ�DG@CKnL >AR�B� x=vFG=%Mqr�DQFQR�B

αi

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NBQJAF

N1, . . . ,Nk

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Z\[AR�FEZ\[AR�FN = a0 +

k∑

i=1

aiNi

D�8��¦8

E[(N −N)2]BQKn@;D�M0@CLA@IM0=%PIR

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����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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888 HjPIK/Z?:

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=

α1

αvar(N)

α2

αvar(N)888

αk

αvar(N)

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E(N2)αi(αi + 1)

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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

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4 x=%FEDQ@IF >AR�B

ai

,

a0 = E(N)(1−k

i=1

ai

αi

α)

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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)

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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

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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

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ai

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: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

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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

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> #¢JALARTPIKn@ XY@IFE@IZ\[APCR�D >ART x=%FQ=%Mqr�DGFER�B(α1 . . . αk)

87vi

2

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BQKnLmD DQKnJAB ���n=%J! ��a1 = . . . = ak = a

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a��JA@ LARN>j�� (R�LA>

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# 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)

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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

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