Particle filter

§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 {Xn }n≥1

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

∼ µ(x) f (x|x ) x x
. Y {Yn }n≥1 {Xn }n≥1 .
{Xn }n≥1 {Yn }n≥1

Yn |(Xn = xn ) ∼ g(yn |xn ) (1.2.2)

· 1 ·

2

n . .
.
(1.2.1) (1.2.2) (HMM)
. .
.
1.2.1. HMM X = {1, . . . , K}

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

(1.2.2) .

1.2.2. X = Rnx Y = Rny X1 ∼ N (0, Σ)

Xn = AXn−1 + BVn ,
Yn = CXn + DWn

Vn ∼ N (0, Inv ) Wn ∼ N (0, Inw ) A B C D
. µ(x) = N (x; 0, Σ) f (x |x = N (x ; Ax, BB T )) g(y|x =
T
N (y ; Cx, DD )). .
.

(1.2.1) (1.2.2) (1.2.1)
{Xn }n≥1 prior distribution (1.2.2) likelihood function.
n
p(x1:n ) = µ(x1 ) f (xk |xk−1 ) (1.2.3)
k=2

n
p(y1:n |x1:n ) = g(yk |xk ) (1.2.4)
k=1

Y1:n = y1:n X1:n
posterior distribution
unnormalised posterior distribution
posterior distribution
p(x1:n , y1:n )
p(x1:n |y1:n ) = (1.2.5)
p(y1:n )
marginal likelihoods

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)

§1.2 Bayesian Inference in Hidden Markov Models 3

(1.2.5) (1.2.6)
( (1.2.3) (1.2.4)). (1.2.7) .
1.2.1 (1.2.7)
(1.2.5)
. 1.2.2 p(x1:n |y1:n )
.
.

p(x1:n |y1:n ) p(y1:n ).
.

• Filtering and Marginal likelihood computation
posterior distribution {p(x1:n |y1:n )}n≥1 marginal likelihoods
{p(y1:n )}n≥1 . p(x1 |y1 ) p(y1 )
p(x1:2 |y1:2 ) p(y1:2 ) .
. marginal distributions
{p(xn |y1:n )}n≥1 {p(x1:n |y1:n )}n≥1 .

• Smoothing: p(x1:T |y1:T )
{p(xn |y1:T )} n = 1, . . . , T .

§1.2.2 Filtering and Marginal Likelihood

.
.
{Y1:n = y1:n } X1:n Xn .
posterior distribution p(x1:n |y1:n ) (1.2.5) prior distribution
p(x1:n ) (1.2.3) likelihood function (1.2.4) . (1.2.5)
unnormalised posterior distribution p(x1:n , y1:n )

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 ) (1.2.8)
= p(x1:n−1 , y1:n−1 )f (xn |xn−1 )g(yn |xn )

4

posterior p(x1:n |y1:n )
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 )
= (1.2.9)
p(y1:n−1 )p(yn |yn−1 )
f (xn |xn−1 )g(yn |xn )
= p(x1:n−1 |y1:n−1 )
p(yn |y1:n−1 )

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 (1.2.10)

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

(1.2.10) .
.
(1.2.9) x1:n−1 marginal distribution p(xn |y1:n )
g(yn |xn )p(xn |y1:n−1 )
p(xn |y1:n ) = (1.2.11)
p(yn |y1:n−1 )

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

(1.2.12) (1.2.11) .
(1.2.9) (1.2.11) (1.2.12).
{p(x1:n |y1:n )} {p(xn |y1:n )}
marginal likelihood p(y1:n )
n
p(y1:n ) = p(y1 ) p(yk |y1:k−1 ) (1.2.13)
k=2

p(yk |y1:k−1 ) (1.2.10) .

§1.2.3 Summary

.
1.2.1 1.2.2 .
.
N .
( N →∞ ).

§1.3 Sequential Monte Carlo Methods 5

§1.3 Sequential Monte Carlo Methods

15 SMC
. SMC
. SMC . SMC
{πn (x1:n )} . πn (x1:n )
n
X .
γn (x1:n )
πn (x1:n ) = (1.3.1)
Zn
γn : X n → R+

Zn = γn (x1:n )dx1:n (1.3.2)

. SMC 1 π1 (x1 ) Z1
2 π2 (x1:2 ) Z2 .
γn (x1:n ) = p(x1:n , y1:n ) Zn = p(y1:n ) πn (x1:n ) =
p(x1:n |y1:n ).

§1.3.1 Basics of Monte Carlo Methods

n πn (x1:n ).
i
N X1:n ∼ πn (x1:n ) πn (x1:n )
N
1
πn (x1:n ) =
ˆ δX1:n (x1:n )
i
N i=1

δx0 (x) x0 Dirac delta mass.
+∞, x = x0
δx0 (x) =
0, x = x0

+∞
δx0 (x)dx = 1.
−∞

πn (xk )
N
1
πn (xk ) =
ˆ δXk (xk )
i
N i=1

ϕn : X n → R

In (ϕn ) = ϕn (x1:n )πn (x1:n )dx1:n

6

N
MC 1 i
In (ϕn ) = ϕn (x1:n )πn (x1:n )dx1:n = ϕn (X1:n )
N i=1

MC
In (ϕn )

MC 1
V In (ϕn ) = ϕ2 (x1:n )πn (x1:n )dx1:n − In (ϕn ) .
n
2
N

N
.
n
X
O(1/N ) .

• 1: πn (x1:n )
.

• 2: πn (x1:n ) n
. πn (x1:n ) n .

§1.3.2 Importance Sampling

IS 1
qn (x1:n )
πn (x1:n ) > 0 ⇒ qn (x1:n ) > 0

(1.3.1) (1.3.2) IS

wn (x1:n )qn (x1:n )
πn (x1:n ) = (1.3.3)
Zn
Zn = wn (x1:n )qn (x1:n )dx1:n (1.3.4)

wn (x1:n )

γn (x1:n )
wn (x1:n ) =
qn (x1:n )

qn (x1:n ) .
i
N X1:n ∼ qn (x1:n ) qn (x1:n )


(1.3.3) (1.3.4)
n
i
πn (x1:n ) = Wn δX1:n (x1:n )
i (1.3.5)
i=1
N
1 i
Zn = wn (X1:n ) (1.3.6)
N i=1

i
i wn (X1:n )
Wn = n j (1.3.7)
j=1 wn (X1:n )

In (ϕn )
N
IS i i
In (ϕn ) = ϕn (x1:n )πn (x1:n )dx1:n = Wn ϕn (X1:n )
i=1

§1.3.3 Sequential Importance Sampling

2 n
.

qn (x1:n ) = qn−1 (x1:n−1 )qn (xn |x1:n−1 )
n
= q1 (x1 ) qk (xk |x1:k−1 ) (1.3.8)
k=2

i
n X1:n ∼ qn (x1:n )
i i i
1 X1 ∼ q1 (x1 ) k = 2, . . . , n Xk ∼ qk (xk |X1:k−1 ).

γn (x1:n )
wn (x1:n ) =
qn (x1:n )
γn−1 (x1:n−1 ) γn (x1:n )
= (1.3.9)
qn−1 (x1:n−1 ) γn−1 (x1:n−1 )qn (xn |x1:n−1 )

wn (x1:n ) = wn−1 (x1:n−1 ) · αn (x1:n )
n
= w1 (x1 ) αk (x1:k ) (1.3.10)
k=2

incremental importance weight αn (x1:n )
γn (x1:n )
αn (x1:n ) = . (1.3.11)
γn−1 (x1:n−1 )qn (xn |x1:n−1 )

8

SIS :
Algorithm 1: Sequential Importance Sampling
1 n=1
2 for i = 1 to N do
i
3 X1 ∼ q1 (x1 )
i i i
4 w1 (X1 ) W1 ∝ w1 (X1 )
5 end
6 for n = 2 to T do
7 for i = 1 to N do
i i
8 Xn ∼ qn (xn |X1:n−1 )
9

i i i
wn (X1:n ) = wn−1 (X1:n−1 ) · αn (X1:n ),
i i
Wn ∝ wn (X1:n ).

10 end
11 end
n πn (x1:n ) Zn πn (x1:n ) (1.3.5) Zn
(1.3.6). Zn /Zn−1
N
Zn i i
= Wn−1 αn X1:n .
Zn−1 i=1

SIS n qn (xn |x1:n−1 ).
wn (x1:n ) .

opt
qn (xn |x1:n−1 ) = πn (xn |x1:n−1 )

x1:n−1 wn (x1:n ) incremental weight

opt γn (x1:n−1 ) γn (x1:n )dxn
αn (x1:n ) = = .
γn−1 (xn−1 ) γn−1 (x1:n−1 )
opt
πn (xn |x1:n−1 ) αn (x1:n ).
opt
qn (xn |x1:n−1 ) .
qn qn (xn |x1:n−1 )
αn (x1:n ) n 2. SIS
. IS n
. SIS IS
(1.3.8) SIS .
.


1.3.1. X =R
n n
πn (x1:n ) = πn (xk ) = N (xk ; 0, 1), (1.3.12)
k=1 k=1
n
x2
k
γn (x1:n ) = exp − ,
k=1
2
Zn = (2π)n/2 .

n n
qn (x1:n ) = qk (xk ) = N (xk ; 0, σ 2 ).
k=1 k=1

1
σ2 > 2
VIS Zn < ∞ relative variance

VIS Zn 1 σ4
n/2
= −1 .
2
Zn N 2σ 2 − 1
1 σ4
σ 2
< σ2 = 1 2σ 2 −1
>1 relative variance
n . σ = 1.2
VIS [Zn ] VIS [Zn ]
qk (xk ) ≈ πn (xk ) N 2
Zn
≈ (1.103)n/2 . n = 1000 N 2
Zn
≈
21 23
1.9 × 10 N ≈ 2 × 10 relative variance
VIS [Zn ]
Z2
= 0.01 .
n

§1.3.4 Resampling

IS SIS n
SMC .
. πn (x1:n ) IS
πn (x1:n ) qn (x1:n )
πn (x1:n ) . πn (x1:n )
i i
IS πn (x1:n ) Wn X1:n
resampling πn (x1:n )
. πn (x1:n ) N
i i
πn (x1:n ) N X1:n Nn
1:N 1 N 1:N
Nn = (Nn , . . . , Nn ) (N, Wn )
1/N . resampled empirical measure
πn (x1:n )
N i
Nn
π n (x1:n ) = δX i (x1:n ) (1.3.13)
i=1
N 1:n

10

i 1:N i
E [Nn |Wn ] = N Wn . π n (x1:n ) πn (x1:n ) .

1
• Systematic Resampling U1 ∼ U 0, N i = 2, . . . , N
i−1 i i−1 k i k
Ui = U1 + N
Nn = Uj : k=1 Wn ≤ Uj ≤ k=1 Wn
0
k=1 = 0.

i i 1:N
• Residual Resampling Nn = N W n N, W n
1:N i i
Nn W n ∝ Wn − N −1 Nn
i i i i
Nn = Nn + N n .

1:N 1:N
• Multinomial Resampling (N, Wn ) Nn .

O(N ) . systematic
resampling
.
πn (x1:n )
In (ϕn ) πn (x1:n )
π n (x1:n ) .
.

.
n n+1
.
. .
§1.3.5 A Generic Sequential Monte Carlo Algorithm

SMC SIS . 1
i i
π1 (x1 ) IS π1 (x1 ) {W1 , X1 }.
.
1 i i
{ N , X 1} . X1
i i j1 j2
N1 N1 j1 = j2 = · · · = jN1
i X1 = X1 =
jN i i
i i
· · · = X1 1
= X1 . SIS X2 ∼ q2 (x2 |X 1 ).
i i
(X 1 , X2 ) π1 (x1 )q2 (x2 |x1 ).
incremental weights α2 (x1:2 ).


. :
Algorithm 2: Sequential Monte Carlo
1 n=1
2 for i = 1 to N do
i
3 X1 ∼ q1 (x1 )
i i i
4 w1 (X1 ) W1 ∝ w1 (X1 )
i i 1 i
5 {W1 , X1 } N { N , X 1}
6 end
7 for n = 2 to T do
8 for i = 1 to N do
i i i i i
9 Xn ∼ qn (xn |X 1:n−1 ) X1:n ← X 1:n−1 , Xn
i i i
10 αn (X1:n ) Wn ∝ αn (X1:n )
i i 1 i
11 {Wn , X1:n } N { N , X 1:n }
12 end
13 end
n πn (x1:n ) . :
N
i
πn (x1:n ) = Wn δX1:n (x1:n )
i (1.3.14)
i=1

N
1 i
π n (x1:n ) = Wn δX i (x1:n ) (1.3.15)
N i=1
1:n

(1.3.14) (1.3.15) . Zn /Zn−1
N
Zn 1 i
= αn X1:n
Zn−1 N i=1

.
.
.
Eﬀective Sample Size (ESS)
. n ESS
N −1
i
ESS = Wn .
i=1

N (
) ESS . ESS 1 N

12

NT . NT = N/2.
i 1 i
. Wn = N
{Wn }
. .
Algorithm 3: Sequential Monte Carlo with Adaptive Resampling
1 n=1
2 for i = 1 to N do
i
3 X1 ∼ q1 (x1 )
i i i
4 w1 (X1 ) W1 ∝ w1 (X1 )
5 if then
i i 1 i
6 {W1 , X1 } N { N , X 1}
i i 1 i
7 {W 1 , X 1 } ← { N , X 1 }
8 else
i i i i
9 {W 1 , X 1 } ← {W1 , X1 }
10 end
11 end
12 for n = 2 to T do
13 for i = 1 to N do
i i i ii
14 Xn ∼ qn (xn |X 1:n−1 ) X1:n ← (X 1:n−1 , Xn
i i i i
15 αn (X1:n ) Wn ∝ W n−1 αn (X1:n )
16 if then
i i 1 i
17 {Wn , X1:n } N { N , X 1:n }
i i 1 i
18 {W n , X n } ← { N , X n }
19 else
i i i i
20 {W 1 , X 1 } ← {Wn , Xn }
21 end
22 end
23 end
πn (x1:n ) .

N
i
πn (x1:n ) = Wn δX1:n (x1:n ),
i (1.3.16)
i=1
N
i
π n (x1:n ) = W n δX i (x1:n ) (1.3.17)
1:n
i=1

n . Zn /Zn−1

N
Zn i i
= W n−1 αn X1:n
Zn−1 i=1

§1.4 Particle Filter 13

1
1.3.1 σ2 > 2
asymptotic variance

VSMC Zn n σ4
1/2
= −1
2
Zn N 2σ 2 − 1

VIS Zn 1 σ4
n/2
= −1 .
2
Zn N 2σ 2 − 1

SMC n IS n
2
. σ = 1.2 qk (xk ) ≈ πn (xk ).
n = 1000 IS N ≈ 2 × 1023
VIS [Zn ] VSMC [Zn ]
Zn 2 = 10−2 . 2
Zn
= 10−2 SMC
N ≈ 104 19 .
§1.3.6 Summary

SMC {πn (x1:n )} {Zn }.

• n qn (xn |x1:n−1 )
αn (x1:n ) n .

• k n > k πn (x1:k )
SMC . n
{πn (x1:n )} SMC . , πn (x1 )
.

§1.4 Particle Filter

SMC SIS
.
{p(x1:n |y1:n )}n≥1 .

ESS .
§1.4.1 SMC for Filtering

SMC {p(x1:n |y1:n )}n≥1

§1.4 Particle Filter 15

Algorithm 4: SMC for Filtering
1 n=1
2 for i = 1 to N do
i
3 X1 ∼ q1 (x1 |y1 )
i µ(xi )g(y1 |X1 )
i
i i
4 w1 (X1 ) = 1
i
q(Xi |y1 )
W1 ∝ w1 (X1 )
i i 1 i
5 {W1 , X1 } N { N , X 1}
6 end
7 for n = 2 to T do
8 for i = 1 to N do
i i i i i
9 Xn ∼ qn (xn |yn , X n−1 ) X1:n ← (X 1:n−1 , Xn )
i i i
i g(yn |Xn )f (Xn |Xn−1 ) i i
10 αn (Xn−1:n ) = i i
q(Xn |yn ,Xn−1 )
Wn ∝ αn (Xn−1:n )
i i 1 i
11 {Wn , X1:n } N { N , X 1:n }
12 end
13 end

[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

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