15
§1.1 Introduction . 15 . 1993 . crude functional approximation. . (MCMC) MCMC. §1.2 Bayesian Inference in Hidden Markov Models §1.2.1 Hidden Markov Models and Inference Aims X {X n } n1 X 1 μ(x 1 ) and X n |(X n-1 = x n-1 ) f (x n |x n-1 ) (1.2.1) μ(x) f (x|x ) x x . Y {Y n } n1 {X n } n1 . {X n } n1 {Y n } n1 Y n |(X n = x n ) g(y n |x n ) (1.2.2) · 1 ·

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Page 1: Particle filter

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1�Ù âfÈÅ�{0�

§1.1 Introduction

G��mÛê��Å�.Jø��4Ù(¹�µe5é�mS�?1ï�"

�,ù�.é¯Käkr��£ãUå§�é¯K�¦)%Ø´é�BµØ�

Ü©4Ù{ü��¹§é�Ü©·�a,��¢S¯K§ÑéJ��¯Kíä�

)Û). �Ù0��“âfÈÅ”�{´�a61��AkÛ�{§å8®²k15c�{¤§§�JÑ�´^5éùJ±¦Ñ)Û)�íä¯Ké�Cq). §gl1993cùa�{�ÄgJѧ§�®²¤�é��5�pd�.?1�`�O��a61�¦)Cq)��{. ÚIO�Cq�{§X61�*Ðk�ùÈÅì?1'�§âfÈÅ�{�Ì�`:´§�Ø�6u?Û�ÛÜ�5zEâ½

?Û� crude functional approximation. ù«Cq¦)¯K�(¹5�¦�p�O�E,Ý", §�XO�Uå�FÃO�§ùa�{®²�^u�«+��

¢�A^§~X§zÆó§!O�ÅÀú!ã²!8I�lÚÅì<�. d§=¦éu@vk¢�5�¦�A^§ùa�{��±^5O�ê��Åó�Ak

Û(MCMC)�{§½ö��|^§�5�OÑ�~k��MCMC�{.

§1.2 Bayesian Inference in Hidden Markov Models

§1.2.1 Hidden Markov Models and Inference Aims

�Ä�� X ��lÑê��ÅL§ {Xn}n≥1 ÷v

X1 ∼ µ(x1) and Xn|(Xn−1 = xn−1) ∼ f(xn|xn−1) (1.2.1)

Ù¥ ∼ L«Ñl©Ù§µ(x) ´��VÇ�ݼê§f(x|x′) L«d x′ £Ä� x �

VÇ�ݼê. ·�a,��´3==®� Y �L§ {Yn}n≥1��OÑ {Xn}n≥1.·�b�§3�½ {Xn}n≥1 �§*� {Yn}n≥1 ´ÚOÕá�§¿�§��>�

©Ù�

Yn|(Xn = xn) ∼ g(yn|xn) (1.2.2)

�{zå�§3ùp·�==�Ä��5��¹§==£Ú*��ݼê��m

· 1 ·

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2 1�Ù âfÈÅ�{0�

eI n Ã'. ��5�¹Ué�*�*Ð����5�¹. �Ù¥b½¤k��.ëêþ�®�.dúª (1.2.1)Ú (1.2.2)£ã��.�¡��Ûê��Å�. (HMM) ½ö

´���G��m�.. ùa�.�)éõa,���.. e¡�ÑA�{ü�~f.~1.2.1. k¡G��m HMMµk X = {1, . . . ,K}§Ïd

Pr(X1 = k) = µ(k), Pr(Xn = k|Xn−1 = l) = f(k|l)

*��.�÷vúª (1.2.2) �?¿�..

~1.2.2. �5pd�.µX = Rnx§Y = Rny§X1 ∼ N (0,Σ)§¿�

Xn = AXn−1 + BVn,

Yn = CXn + DWn

Ù¥ Vn ∼ N (0, Inv)§Wn ∼ N (0, Inw

)§A§B§C§D ´äk·��ê�Ý

. 3ù«�¹e µ(x) = N (x; 0,Σ)§f(x′|x = N (x′;Ax,BBT )) Ú g(y|x =N (y′;Cx,DDT )). ù«�.e§é¯K�íä´�±¦Ñ)Û)�. §�2�¦^38I�lÚ&Ò?n+�.

úª (1.2.1) Ú (1.2.2) ½Â����d�.§Ù¥úª (1.2.1) ½ÂL§ {Xn}n≥1 � prior distribution§úª (1.2.2) ½Â likelihood function. =µ

p(x1:n) = µ(x1)n∏

k=2

f(xk|xk−1) (1.2.3)

Ú

p(y1:n|x1:n) =n∏

k=1

g(yk|xk) (1.2.4)

3ù������dµee§��½*�S������ Y1:n = y1:n§é X1:n

�íä�6u posterior distribution

posterior distribution︷ ︸︸ ︷p(x1:n|y1:n) =

unnormalised posterior distribution︷ ︸︸ ︷p(x1:n, y1:n)

p(y1:n)︸ ︷︷ ︸marginal likelihoods

(1.2.5)

Ù¥

p(x1:n, y1:n) = p(x1:n)p(y1:n|x1:n) (1.2.6)

p(y1:n) =∫

p(x1:n, y1:n)dx1:n (1.2.7)

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§1.2 Bayesian Inference in Hidden Markov Models 3

��§éuúª (1.2.5) L«���©Ù§©fdúª (1.2.6) �±N´�O�Ñ5 (�\ (1.2.3) Ú (1.2.4)). '�´úª (1.2.7) L«�©1Ø�BO�.

éu~ 1.2.1 ¥?Ø�k¡G��mÛê��Å�.§úª (1.2.7) ¥�È©�±ÏLk¡�¦Úö�5�¤§Ïdúª (1.2.5) ¥���©Ù�±°(/O�Ñ5. éu~ 1.2.2 ¥?Ø��5pd�.§éN´����©Ù p(x1:n|y1:n)�´��pd©Ù§§�þ�Ú��Udk�ùÈÅ�{¦). , §éu�õê��5�pd��.§Ø�U¦)��©Ù�4)§ÏdI�|^ê���{.âfÈÅ�{´�«(¹ qr��Äu�ý��{§UJøCqÑl��©Ù

p(x1:n|y1:n) ���§l UCq¦) p(y1:n). âfÈÅ�{´S��AkÛ�{��f8.

âfÈÅ�{�±^5¦)e¡�¯Kµ

• Filtering and Marginal likelihood computationµb½·�a,��´S�/Cq posterior distribution {p(x1:n|y1:n)}n≥1 Ú marginal likelihoods{p(y1:n)}n≥1. �=§·�F"31����:Cq p(x1|y1)Ú p(y1)§31����:Cq p(x1:2|y1:2) Ú p(y1:2)§�daí. ·�òrù�¯K¡���`ÈůK. 3�õê©z¥§ÈůK��´�O marginal distributions{p(xn|y1:n)}n≥1 Ø´éÜ©Ù {p(x1:n|y1:n)}n≥1.

• Smoothing: b½l��éÜ©Ù p(x1:T |y1:T ) æ�§F"Cq/��>�©Ù {p(xn|y1:T )}§Ù¥ n = 1, . . . , T .

§1.2.2 Filtering and Marginal Likelihood

·�a,��§�=�õê'uâfÈÅ�©z¤ïÄ�¯K§Ò´Èů

Kµ�½��c��¤Â8��¤k*�&E§�OÛê��Å�.�c���

G��©Ù. ÈÅk�ÿ���´�½��c��¤Â8��¤k*�&E§�OÛê��Å�.��c���¤kG�S�. /ªz£ã§=��½��*�S� {Y1:n = y1:n}§íäÑ X1:n �©Ù½ö´ Xn �©Ù.

posterior distribution p(x1:n|y1:n) dúª (1.2.5) ½Â§prior distributionp(x1:n)dúª (1.2.3)½Â§likelihood functiondúª (1.2.4)½Â.úª (1.2.5)¥�©f§= unnormalised posterior distribution p(x1:n, y1:n) ÷v

p(x1:n, y1:n) = p(x1:n−1, y1:n−1)p(xn, yn|x1:n−1, y1:n−1)

= p(x1:n−1, y1:n−1)p(xn|xn−1)p(yn|xn)

= p(x1:n−1, y1:n−1)f(xn|xn−1)g(yn|xn)

(1.2.8)

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4 1�Ù âfÈÅ�{0�

Ïd posterior p(x1:n|y1:n) ÷ve¡�48ªf

p(x1:n|y1:n) =p(x1:n, y1:n)

p1:n

=p(x1:n−1, y1:n−1)f(xn|xn−1)g(yn|xn)

p(y1:n−1)p(yn|yn−1)

= p(x1:n−1|y1:n−1)f(xn|xn−1)g(yn|xn)

p(yn|y1:n−1)

(1.2.9)

Ù¥

p(yn|y1:n−1) =∫

p(yn, xn−1:n|y1:n−1)dxn−1:n

=∫

p(xn−1|y1:n−1)p(yn, xn|xn−1, y1:n−1)dxn−1:n

=∫

p(xn−1|y1:n−1)f(xn|xn−1)g(yn|xn)dxn−1:n

(1.2.10)

é��©Ù�O��ª�´�6uúª (1.2.10) ¥�¦È©. Ï�8céù�È©ªfØЦ§¤±��©Ù�éJ¦Ñ5.éúª (1.2.9)§È©K x1:n−1§Ò�±�� marginal distribution p(xn|y1:n)

p(xn|y1:n) =g(yn|xn)p(xn|y1:n−1)

p(yn|y1:n−1)(1.2.11)

Ù¥

p(xn|y1:n−1) =∫

f(xn|xn−1)p(xn−1|y1:n−1)dxn−1 (1.2.12)

úª (1.2.12) �¡�ýÿ§úª (1.2.11) �¡��#. , �õêâfÈÅ�{�6u48ª (1.2.9) �ê�Cq§ Ø´úª (1.2.11) Ú (1.2.12).XJ·�UO�Ñ {p(x1:n|y1:n)}§ÏdUS�/O�Ñ {p(xn|y1:n)}§?

marginal likelihood p(y1:n) �U�48/O�

p(y1:n) = p(y1)n∏

k=2

p(yk|y1:k−1) (1.2.13)

Ù¥ p(yk|y1:k−1) äkúª (1.2.10) �/ª.

§1.2.3 Summary

é��5�pdÄ��.���díä�6u��©ÙS�Ú§�>�©Ù.Ø{ü�¯K§X~ 1.2.1 Ú 1.2.2 §Ø�U��ù©Ù�4). ùp·�ò?Øù©Ù��AkÛCq. �AkÛCq�{´�aê��{§�Cq�©Ùd N ��Å���§¡�âf§5Cq. ù�{�Ì�`:´3éf�b�e§§�UJøé8I©ÙìC��O (~X� N →∞ �).

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§1.3 Sequential Monte Carlo Methods 5

§1.3 Sequential Monte Carlo Methods

3L�15c��mp§^âf��{5?1ÈÅÚ²w¤� SMC �{A^�²~�~f. ¢Sþ§3�õê©z¥§râfÈÅÚ SMC ��Ó��Vg. ùp·�rN SMC ¢Sþ�¹����2. SMC �{´�AkÛ�{��a§§S�/l8IVÇ�ÝS� {πn(x1:n)} ¥æ�. z��©Ù πn(x1:n) ½Â3(k��m X n þ. L«�

πn(x1:n) =γn(x1:n)

Zn

(1.3.1)

·�==�¦ γn : X n → R+ ´®��§8�z~þ

Zn =∫

γn(x1:n)dx1:n (1.3.2)

�±´���. SMC Jø�� 1 � π1(x1) ���CqÚ Z1 ����O§,

�3�� 2§Jø π2(x1:2) ���CqÚ Z2 ����O§�daí. ~X§éuÈůK§·��±k γn(x1:n) = p(x1:n, y1:n)§Zn = p(y1:n)§Ïd πn(x1:n) =p(x1:n|y1:n).

§1.3.1 Basics of Monte Carlo Methods

Äk�ÄCq����� n ��½��VÇ�Ý πn(x1:n). XJ·�l§æ�N �Õá��ÅCþ§X i

1:n ∼ πn(x1:n)§Ò�±^�AkÛ�{5Cq πn(x1:n)

πn(x1:n) =1N

N∑i=1

δXi1:n

(x1:n)

Ù¥ δx0(x) L«3 x0 ?� Dirac delta mass. =

δx0(x) ={

+∞, x = x0

0, x 6= x0

�Ó�÷v ∫ +∞

−∞δx0(x)dx = 1.

Äuù�Cq§�±é�B/Cq?Û�>� πn(xk)

πn(xk) =1N

N∑i=1

δXik(xk)

?Û¼ê ϕn : X n → R �Ï"

In(ϕn) =∫

ϕn(x1:n)πn(x1:n)dx1:n

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6 1�Ù âfÈÅ�{0�

�±��O¤

IMCn (ϕn) =

∫ϕn(x1:n)πn(x1:n)dx1:n =

1N

N∑i=1

ϕn(X i1:n)

5¿�Oþ IMCn (ϕn) E,´���ÅCþ§éN´�yù��Oþ´Ã �O§

¿�§����

V[IMC

n (ϕn)]

=1N

(∫ϕ2

n(x1:n)πn(x1:n)dx1:n − I2n(ϕn)

).

Ä���AkÛ�{�Ì�g�´§XJ·�®�k N ���ægu��©Ù§

K�±^ù��5CqT©Ù§±9½Â3�ÅCþþ�¼ê�êÆÏ". �'uIO�Cq�{§�AkÛ�{�Ì�`:�O�����m X n ��êÃ'§

´± O(1/N) ��Çeü. �§kü�Ì��¯Kµ

• ¯K1: XJ πn(x1:n) ´��E,�p�VÇ©Ù§·�Ø�U��l§æ�.

• ¯K2: =¦��XÛ°(/l πn(x1:n) æ�§æ��O�E,5��´ n

��5�. �S�/l πn(x1:n) æ��§O�E,ݬ�X n �O\ O\.

§1.3.2 Importance Sampling

IS �{�±^5)û¯K1§§�Ø%g�´Ú\��­�5©Ù½JÆ©

Ù qn(x1:n)§÷v

πn(x1:n) > 0⇒ qn(x1:n) > 0

3ù«�¹e§lúª (1.3.1) Ú (1.3.2) �� IS �Ý

πn(x1:n) =wn(x1:n)qn(x1:n)

Zn

(1.3.3)

Zn =∫

wn(x1:n)qn(x1:n)dx1:n (1.3.4)

Ù¥ wn(x1:n) ´�8�z��­¼ê

wn(x1:n) =γn(x1:n)qn(x1:n)

AO/§·��±ÀJN´æ��­�5©Ù qn(x1:n)§~X��õ�pd©Ù.b½·�æ� N �Õá��� X i

1:n ∼ qn(x1:n)§,�ò qn(x1:n) ��AkÛ

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§1.3 Sequential Monte Carlo Methods 7

Cq�\��§ (1.3.3) Ú (1.3.4) ¥§��

πn(x1:n) =n∑

i=1

W inδXi

1:n(x1:n) (1.3.5)

Zn =1N

N∑i=1

wn(X i1:n) (1.3.6)

Ù¥

W in =

wn(X i1:n)∑n

j=1 wn(Xj1:n)

(1.3.7)

·�a,��´O� In(ϕn)§�±¦^Xe��O

IISn (ϕn) =

∫ϕn(x1:n)πn(x1:n)dx1:n =

N∑i=1

W inϕn(X i

1:n)

§1.3.3 Sequential Importance Sampling

e¡5)û¯K2§=é��«)§¦�éu?¿��� n §æ���mE,

Ý´�½�. ¦)g´´�ÀJ��äkXe(��­�5©Ù

qn(x1:n) = qn−1(x1:n−1)qn(xn|x1:n−1)

= q1(x1)n∏

k=2

qk(xk|x1:k−1) (1.3.8)

þ¡�úª¿�X§XJ·�3�� n§�æ�âf X i1:n ∼ qn(x1:n)§�±3��

1 �§æ� X i1 ∼ q1(x1)§,�éu k = 2, . . . , n �§æ� X i

k ∼ qk(xk|X i1:k−1).

¦^e¡�©)

wn(x1:n) =γn(x1:n)qn(x1:n)

=γn−1(x1:n−1)qn−1(x1:n−1)

γn(x1:n)γn−1(x1:n−1)qn(xn|x1:n−1)

(1.3.9)

�âféA��8�z��­U�48/O�

wn(x1:n) = wn−1(x1:n−1) · αn(x1:n)

= w1(x1)n∏

k=2

αk(x1:k) (1.3.10)

Ù¥ incremental importance weight ¼ê αn(x1:n) �

αn(x1:n) =γn(x1:n)

γn−1(x1:n−1)qn(xn|x1:n−1). (1.3.11)

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8 1�Ù âfÈÅ�{0�

SIS�{Xe:Algorithm 1: Sequential Importance Sampling

3�� n = 11

for i = 1 to N do2

æ� X i1 ∼ q1(x1)3

O��­ w1(X i1) Ú W i

1 ∝ w1(X i1)4

end5

for n = 2 to T do6

for i = 1 to N do7

æ� X in ∼ qn(xn|X i

1:n−1)8

O��­9

wn(X i1:n) = wn−1(X i

1:n−1) · αn(X i1:n),

W in ∝ wn(X i

1:n).

end10

end11

éu?Û�� n§�{©O�� πn(x1:n) Ú Zn ��O πn(x1:n) (1.3.5)Ú Zn

(1.3.6). dù��8Ü�U�� Zn/Zn−1 ��O

Zn

Zn−1

=N∑

i=1

W in−1αn

(X i

1:n

).

3SISµee§3�� n§ r���gd´ÀJ qn(xn|x1:n−1). Ün�ÀJüÑ´¦� wn(x1:n) �����z. �ÀJ

qoptn (xn|x1:n−1) = πn(xn|x1:n−1)

�§3 x1:n−1 �^�e§wn(x1:n) ����"§¿��A� incremental weight�

αoptn (x1:n) =

γn(x1:n−1)γn−1(xn−1)

=∫

γn(x1:n)dxn

γn−1(x1:n−1).

, §·�Ø�Ul πn(xn|x1:n−1) æ�§�Ø�UO�Ñ αoptn (x1:n). Ïd·�

I�|^ qoptn (xn|x1:n−1) ���Cq.

�Ün�ÀJ qn ¿�^r'%�´~XÈÅùa¯K�§l qn(xn|x1:n−1)æ�ÚO� αn(x1:n) ¤I���m´� n Ã'�§l )û¯K2. , SIS�{k��":. =¦´éuIO� IS �{§����O����X n ¤�ê?

�O�. Ï� SIS ==´ IS ���A~§·�==��­�5©Ùäkúª(1.3.8) �/ª§Ï SIS �3Ó��¯K. ·�ÏL��{ü�~f5�ãù�¯K.

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§1.3 Sequential Monte Carlo Methods 9

~1.3.1. �Ä��~f§X = R§¿�

πn(x1:n) =n∏

k=1

πn(xk) =n∏

k=1

N (xk; 0, 1), (1.3.12)

γn(x1:n) =n∏

k=1

exp(−x2

k

2

),

Zn = (2π)n/2.

ÀJ­�5©Ù

qn(x1:n) =n∏

k=1

qk(xk) =n∏

k=1

N (xk; 0, σ2).

� σ2 > 12�§VIS

[Zn

]<∞§¿� relative variance

VIS

[Zn

]Z2

n

=1N

[(σ4

2σ2 − 1

)n/2

− 1

].

N´�y§éu?¿� σ ÷v 12

< σ2 6= 1§k σ4

2σ2−1> 1§d� relative variance

�X n�êO\. ~X§XJ·�ÀJ σ = 1.2§�,d�U����Ð�­�5©Ù qk(xk) ≈ πn(xk)§�´ N

VIS[Zn]Z2

n≈ (1.103)n/2. � n = 1000�§N

VIS[Zn]Z2

n≈

1.9 × 1021§d�§·�I�¦^ N ≈ 2 × 1023 �âf5¦� relative varianceVIS[Zn]

Z2n

= 0.01§ùA�´Ã{���.

§1.3.4 Resampling

dc¡�Qã��§IS Ú SIS �{Jø��O����X n �êO\§­æ

�Eâ´ SMC �{¥���'�Ú½§5)ûù��¯K. ­æ�´��é�*��{§äk­��¢SÚnØd�. Äk�Äé8I©Ù πn(x1:n) ��� IS Cq πn(x1:n)§ù�Cq´Äul©Ù qn(x1:n) æ������§ vkJøCqægu8I©Ù πn(x1:n) ���. ���ægu πn(x1:n) �Cq��§·��±l§� IS Cq πn(x1:n) ?1æ�§=·�±VÇ W i

n ÀJ X i1:n§ù�ö�¡

�� resampling§Ï�§´l����Ò´²Læ� ���Cq©Ù πn(x1:n)¥?1æ�. XJ·�él πn(x1:n) �� N ���a,�§K·��±{ü/

l πn(x1:n) ­æ� N g§ù�du4z��âf X i1:n �� N i

n ���§�¦÷

vXe�ªµN 1:Nn = (N 1

n, . . . , NNn ) Ñl��ëê�þ� (N,W 1:N

n ) �õ�ª©Ù§¿��z����D� 1/N . ·�^ resampled empirical measure 5Cqπn(x1:n)

πn(x1:n) =N∑

i=1

N in

NδXi

1:n(x1:n) (1.3.13)

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10 1�Ù âfÈÅ�{0�

Ù¥ E [N in|W 1:N

n ] = NW in. Ïd πn(x1:n) ´ πn(x1:n) ���àCq.

©z¥JÑn«­æ��{µ

• Systematic Resampling æ� U1 ∼ U[0, 1

N

]§éu¤k� i = 2, . . . , N§

½Â Ui = U1 + i−1N§,��½ N i

n =∣∣∣{Uj :

∑i−1

k=1 W kn ≤ Uj ≤

∑i

k=1 W kn

}∣∣∣§Ù¥�½

∑0

k=1 = 0.

• Residual Resampling � N in = bNW i

nc§l��õ�ª©Ù(N,W

1:N

n

)¥æ� N

1:N

n §Ù¥ Wi

n ∝W in −N−1N i

n§,�� N in = N i

n + Ni

n.

• Multinomial Resampling l����ª©Ù (N,W 1:Nn ) ¥æ� N 1:N

n .

3 O(N) �mE,ÝS§�±k�/l��õ�ª©Ùæ�. ,¡ systematicresampling �{ÏÙ4´¢y§ ©z¥¦^�2���«�{§¿�3�õê

A^|Üe§Ù5U��LÙ§�æ��{.

­æ�#N·���Cqægu πn(x1:n) ���§�7L�Ù�¿£�§XJ·�a,��´�O In(ϕn)§K¦^ πn(x1:n) ?1�O§�Oþ���'¦^πn(x1:n) ����Oþ�����. ÏL­æ�§·�¢SþO\���D(. , §­æ����²w�`:´§#N·�±�p�VÇ£Ø$�­�@âf§ù´4Ùk^�§Ï�3·�a,��S�µee§·�¿ØI�D4@

$VÇ�âf§ ´rO�å8¥3pVÇ�@âfþ¡. 7L�Ù§k�U¬Ñy§��� n ��$�­�âf§3�� n + 1 �U¬k�p��­§3ù«�¹e§­æ�ÒL¤. , éu·��ca,���O¯K ó§­æ�®²�y²´k��. �*þ`§­æ�±O\����d5¦XÚ3ò5�­½.

§1.3.5 A Generic Sequential Monte Carlo Algorithm

SMC �{¢�þÒ´ò SIS Ú­æ�éÜå5. 3�� 1§·�O�Ñπ1(x1) � IS Cq π1(x1)µ��D��âf8Ü {W i

1, Xi1}. ,�·�¦^­æ�

Ú½±�p�VÇ�Ø@$�­�âf§Ó�E�@p�­�âf. ·�^{ 1

N, X

i

1} L«²L­æ����­�âf8Ü. P4§z���k�âf X i1 k

N i1 ���§Ïd�3 N i

1 �ØÓ�eI j1 6= j2 6= · · · 6= jNi1¦� X

j1

1 = Xj2

1 =

· · · = XjNi

11 = X i

1. ­æ���§·�¦^ SIS �{æ� X i2 ∼ q2(x2|X

i

1). Ïd§(X

i

1, Xi2) Cq/Ñl©Ù π1(x1)q2(x2|x1). d�§�A�­�5�­Ò�u

incremental weights α2(x1:2). ,�§­æ�ùvk8�z�­�âf§�da

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§1.3 Sequential Monte Carlo Methods 11

í. ����{Xe:Algorithm 2: Sequential Monte Carlo

3�� n = 11

for i = 1 to N do2

æ� X i1 ∼ q1(x1)3

O��­ w1(X i1) Ú W i

1 ∝ w1(X i1)4

­æ� {W i1, X

i1}§�� N ���­�âf { 1

N, X

i

1}5

end6

for n = 2 to T do7

for i = 1 to N do8

æ� X in ∼ qn(xn|X

i

1:n−1) ¿�� X i1:n ←

(X

i

1:n−1, Xin

)9

O��­ αn(X i1:n) ¿� W i

n ∝ αn(X i1:n)10

­æ� {W in, X i

1:n}§�� N ���­�âf { 1N

, Xi

1:n}11

end12

end13

é?Û�� n§�{�� πn(x1:n) �ü�Cq. æ����Cq:

πn(x1:n) =N∑

i=1

W inδXi

1:n(x1:n) (1.3.14)

Ú­æ����Cqµ

πn(x1:n) =1N

N∑i=1

W inδ

Xi1:n

(x1:n) (1.3.15)

Cq (1.3.14) 'Cq (1.3.15) �Ð. �{��� Zn/Zn−1 ���Cq

Zn

Zn−1

=1N

N∑i=1

αn

(X i

1:n

)�Xc¡®²J��§­æ�U�ØK$�­�âf¿�E�p�­�âf§ù

´±Ú\������d�. XJvk­æ��c§vk8�z��­�k�����§K���­æ�Ú½´v7��. Ïd§3¢S�¥§�Ün�üÑ´§�vk²L8�z��­���pu�½�K��§â?1­æ�. ÏL*d¤¢� Effective Sample Size (ESS) IO½Â��­��C55�ä´Ä��­æ��^�. 3�� n � ESS ½Â�

ESS =

(N∑

i=1

(W i

n

))−1

.

�±�ù��)ºµÄu N �D����?1íä�±Cq�d (l�O����Ý) uÄul8I©ÙæÑ� ESS �IÐ����íä. ESS ��3 1 � N

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Mó�Ü�²

12 1�Ù âfÈÅ�{0�

�m§==�§$u�½�K� NT �â?1­æ�. Ï~ NT = N/2. ��±r�­�����ä´ÄI�­æ��IO. � W i

n = 1N�§�­ {W i

n} �����. ���u��K��§?1­æ�.

Algorithm 3: Sequential Monte Carlo with Adaptive Resampling

3�� n = 11

for i = 1 to N do2

æ� X i1 ∼ q1(x1)3

O��­ w1(X i1) Ú W i

1 ∝ w1(X i1)4

if æ�IO then5

­æ� {W i1, X

i1}§�� N ���­�âf { 1

N, X

i

1}6

� {W i

1, Xi

1} ← { 1N

, Xi

1}7

else8

� {W i

1, Xi

1} ← {W i1, X

i1}9

end10

end11

for n = 2 to T do12

for i = 1 to N do13

æ� X in ∼ qn(xn|X

i

1:n−1) ¿�� X i1:n ← (X

i

1:n−1, Xin14

O��­ αn(X i1:n) ¿� W i

n ∝Wi

n−1αn(X i1:n)15

if æ�IO then16

­æ� {W in, X i

1:n}§�� N ���­�âf { 1N

, Xi

1:n}17

� {W i

n, Xi

n} ← { 1N

, Xi

n}18

else19

� {W i

1, Xi

1} ← {W in, X i

n}20

end21

end22

end23

�{�� πn(x1:n) �ü�Cq.

πn(x1:n) =N∑

i=1

W inδXi

1:n(x1:n), (1.3.16)

πn(x1:n) =N∑

i=1

Wi

nδX

i1:n

(x1:n) (1.3.17)

XJ3�� n vk?1­æ�§ùü�ªf´���. �{��� Zn/Zn−1 ��

�Cq

Zn

Zn−1

=N∑

i=1

Wi

n−1αn

(X i

1:n

)

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Mó�Ü�²

§1.4 Particle Filter 13

UY�Ä~f 1.3.1§� σ2 > 12�§k asymptotic variance

VSMC

[Zn

]Z2

n

=n

N

[(σ4

2σ2 − 1

)1/2

− 1

]

VIS

[Zn

]Z2

n

=1N

[(σ4

2σ2 − 1

)n/2

− 1

].

?1'�§SMC�O�ìC���X n�5O\§ IS�O����X n�êO

\. ~X§XJÀJ σ2 = 1.2§K�±����Ð�­�5©Ù qk(xk) ≈ πn(xk).3ù«�¹e§� n = 1000 �§IS �{7L¦^ N ≈ 2 × 1023 �âf5��VIS[Zn]

Z2n

= 10−2. , ���Ó��5U§VSMC[Zn]

Z2n

= 10−2§SMC �{==I�N ≈ 104 �âf§�� 19 ��U?.

§1.3.6 Summary

JÑ����� SMC �{§ 5S�/Cq {πn(x1:n)} Ú {Zn}.

• ÃØ3�o�¹e§Ñ�±Õáu�m n§l©Ù qn(xn|x1:n−1) ?1æ�¿��O αn(x1:n)§�{��mE,Ýج�X n �O\ O\.

• é?Û� k§�3X n > k duëY�­æ�Ú½§¬¦�é πn(x1:k) �SMC Cq==d��üÕ�âf�¤. Ïd� n ���§Ø�U��éÜ©

Ù {πn(x1:n)} ���Ð� SMC Cq. ù3¢S�¥,ÏL*Cq πn(x1)�ØÓâf��ê�±éN´�uyù�:.

§1.4 Particle Filter

c¡J� SMC �{¢�þÒ´ SIS Ú­æ��{�(ܧ �«|^�Å

��é©Ù½½Â3©Ùþ�¼ê�êÆÏ"?1�O��{. ��ÄÈůK�§·�F"S�/O���©Ù {p(x1:n|y1:n)}n≥1 �ê�Cq. �L«�{ü§e¡JÑ��{3z���Ñ?1­æ�§ ¢S¥§�í���ª´�

ESS �u��K��§âI�?1­æ�.

§1.4.1 SMC for Filtering

�ò SMC �{A^�Èŵee§=O�Ñ©Ù {p(x1:n|y1:n)}n≥1 �ê�

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Mó�Ü�²

14 1�Ù âfÈÅ�{0�

Cq§�I�ÀJµ

πn(x1:n) = p(x1:n|y1:n)

γ(x1:n) = p(x1:n, y1:n)

Zn = p(y1:n) (1.4.1)

nonumber (1.4.2)

éu­�5©Ù§·�v7�ÀJ qn(x1:n)§ �I�ÀJ qn(xn|x1:n−1). Ï�Xc¤ã§3 IS �{p§·�ÀJ�­�©Ù´ qn(x1:n) ù«/ª§ �3 SIS �{p§·��I�­�5©Ùäk qn(xn|x1:n−1) ù«/ªÒ. XJ���z�� n ���5�­���§·�ATÀJ�`�­�5©Ùµ

qoptn (xn|x1:n−1) = πn(xn|x1:n−1Z)

= p(xn|yn, xn−1)

=g(yn|xn)f(xn|xn−1)

p(yn|xn−1)(1.4.3)

éA� incremental importance weight ´

αn(x1:n) = p(yn|xn−1)

3¢S¥§Ø�Ul qoptn (xn|x1:n−1) ?1æ�§ ´ATÀJXe/ª�­�5

©Ùµ

qn(xn|x1:n−1) = q(xn|yn, xn−1) (1.4.4)

éÜúª(1.3.9)§(1.3.11) Ú (1.4.4)§�±�Ñ incremental weight

αn(x1:n) = αn(xn−1:n) =g(yn|xn)f(xn|xn−1)

q(xn|yn, xn−1).

Page 15: Particle filter

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§1.4 Particle Filter 15

Algorithm 4: SMC for Filtering

3�� n = 11

for i = 1 to N do2

æ� X i1 ∼ q1(x1|y1)3

O��­ w1(X i1) = µ(xi

1)g(y1|Xi1)

q(Xii |y1)

Ú W i1 ∝ w1(X i

1)4

­æ� {W i1, X

i1}§�� N ���­�âf { 1

N, X

i

1}5

end6

for n = 2 to T do7

for i = 1 to N do8

æ� X in ∼ qn(xn|yn, X

i

n−1) ¿�� X i1:n ← (X

i

1:n−1, Xin)9

O��­ αn(X in−1:n) = g(yn|Xi

n)f(Xin|X

in−1)

q(Xin|yn,Xi

n−1)¿� W i

n ∝ αn(X in−1:n)10

­æ� {W in, X i

1:n}§�� N ���­�âf { 1N

, Xi

1:n}11

end12

end13

�z

[1] A.D. and A. Johansen, Particle filtering and smoothing: Fifteen years later, in Hand-

book of Nonlinear Filtering (eds. D. Crisan et B. Rozovsky), Oxford University Press,

2009. See http://www.cs.ubc.ca/~arnaud