Sampling strategies for Sequential Monte Carlo (SMC) methods

Sampling strategies for
Sequential Monte Carlo methods
Arnaud Doucet1
, Stéphane Sénécal2
1
Department of Engineering, University of Cambridge
2
The Institute of Statistical Mathematics
2004
thanks to the Japanese Ministry of Education and the Japan Society for
the Promotion of Science
1

Overview
– Introduction : state space models, Monte Carlo methods
– Sequential Importance Sampling/Resampling
– Strategies for sampling
– Examples, applications
– References
2

Estimation of state space models
xt = ft(xt−1, ut) yt = gt(xt, vt)
p(x0:t|y1:t) → p(xt|y1:t) = p(x0:t|y1:t)dx0:t−1
distribution of x0:t ⇒ computation of estimate x0:t :
x0:t = x0:tp(x0:t|y1:t)dx0:t → Ep(.|y1:t){f(x0:t)}
x0:t = arg max
x0:t
p(x0:t|y1:t)
3

Computation of the estimates
p(x0:t|y1:t) ⇒ multidimensionnal, non-standard distributions :
→ analytical, numerical approximations
→ integration, optimisation methods
⇒ Monte Carlo techniques
4

Monte Carlo approach
compute estimates for distribution π(.) → samples x1, . . . , xN ∼ π
x
pi(x)
x_1 x_N
⇒ distribution πN = 1
N
N
i=1 δxi approximates π(.)
5

Monte Carlo estimates
SN (f) =
1
N
N
i=1
f(xi) −→ f(x)π(x)dx = Eπ{f(x)}
arg max(xi)1≤i≤N
πN (xi) approximates arg maxx π(x)
⇒ sampling xi ∼ π diﬃcult
→ importance sampling techniques
6

Importance Sampling
xi ∼ π → candidate/proposal distribution xi ∼ g
x
g(x)
pi(x)
x_Nx_1
7

Importance Sampling
xi ∼ g = π → (xi, wi) weighted sample
⇒ weight wi =
π(xi)
g(xi)
x
g(x)
pi(x)
x_Nx_1
8

Estimation
importance sampling → computation of Monte Carlo estimates
e. g. expectations Eπ{f(x)} :
f(x)
π(x)
g(x)
g(x)dx = f(x)π(x)dx
N
i=1
wif(xi) → f(x)π(x)dx = Eπ{f(x)}
dynamic model (xt, yt) ⇒ recursive estimation x0:t−1 → x0:t
Monte Carlo techniques ⇒ sampling sequences x
(i)
0:t−1 → x
(i)
0:t
9

Sequential simulation
sampling sequences x
(i)
0:t ∼ πt(x0:t) recursively :
time
variable
state
x
p(x,t) target distribution:
t
t2
t1
p(x,t2)
x_t1
x_t2
p(x_t1)
p(x_t2)
p(x,t1)
10

Sequential simulation : importance sampling
samples x
(i)
0:t ∼ πt(x0:t) approximated by weighted particles
(x
(i)
0:t, w
(i)
t )1≤i≤N
time
p(x,t2)
t
t2
t1
x
p(x,t1)
11

Sequential importance sampling
diﬀusing particles x
(i)
0:t1
→ x
(i)
0:t2
time
p(x,t2)
t
x
p(x,t1)
t2
t1
⇒ sampling scheme x
(i)
0:t−1 → x
(i)
0:t
12

Sequential importance sampling
updating weights w
(i)
t1
→ w
(i)
t2
time
p(x,t2)
t
p(x,t1)
x
t2
t1
⇒ updating rule w
(i)
t−1 → w
(i)
t
13

Sequential Importance Sampling
x0:t ∼ πt(x0:t) ⇒ (x
(i)
0:t, w
(i)
t )1≤i≤N
Simulation scheme t − 1 → t :
– Sampling step x
(i)
t ∼ qt(xt|x
(i)
0:t−1)
– Updating weights
w
(i)
t ∝ w
(i)
t−1 ×
πt(x
(i)
0:t−1, x
(i)
t )
πt−1(x
(i)
0:t−1)qt(x
(i)
t |x
(i)
0:t−1)
incremental weight (iw)
normalizing
N
i=1 w
(i)
t = 1
14

x0:t ∼ πt(x0:t) ⇒ (x
(i)
0:t, w
(i)
t )1≤i≤N
proposal + reweighting →
pi(x_t)
x_t
15

proposal + reweighting → var{(w
(i)
t )1≤i≤N } with t
x_t
pi(x_t)
→ w
(i)
t ≈ 0 for all i except one
16

⇒ Resampling
x_t
pi(x_t)
0 x_t^(1)
x_t^(j)1x_t^(i)2 x_t^(k)3
x_t^(N)0
→ draw N particles paths from the set (x
(i)
0:t)1≤i≤N
with probability (w
(i)
t )1≤i≤N
17

Sequential Importance Sampling/Resampling
Simulation scheme t − 1 → t :
– Sampling step x
,(i)
t ∼ qt(x,
t|x
(i)
0:t−1)
– Updating weights w
(i)
t ∝ w
(i)
t−1 ×
πt(x
(i)
0:t−1,x
,(i)
t )
πt−1(x
(i)
0:t−1)qt(x
,(i)
t |x
(i)
0:t−1)
→ parallel computing
– ⇒ Resampling step : sample N paths from (x
(i)
0:t−1, x
,(i)
t )1≤i≤N
→ particles interacting : computation at least O(N)
18

SISR for recursive estimation of state space models
xt = ft(xt−1, ut) → p(xt|xt−1)
yt = gt(xt, vt) → p(yt|xt)
Usual SISR : Bootstrap filter (Gordon et al. 93, Kitagawa 96) :
– Sampling step x
(i)
t ∼ p(xt|x
(i)
t−1)
– Updating weights : incremental weight w
(i)
t ∝ w
(i)
t−1 × iw
iw ∝ p(yt|x
(i)
t )
– Stratified/Deterministic resampling
efficient, easy, fast for a wide class of models
→ tracking, time series
19

Overview - Break
– Introduction :
→ state space models
→ estimation, computating estimates via Monte Carlo methods
→ importance sampling
– recursive estimation → sequential simulation
⇒ Sequential Importance Sampling/Resampling
– ⇒ Strategies for sampling :
→ designing/sampling “optimal” candidate distribution
→ considering blocks of variables : reweighting, → sampling
– Examples and applications
20

Improving simulation
sampling multimodal, multidimensional distributions
model with informative observation → peaky likelihood
→ prior dynamics to diffuse particles : poor approximation results
→ efficient propagation for a finite number of particles N
⇒ need for good sampling proposals
21

Approximations
Sampling the predictive distribution πt(xt|x0:t−1) = p(xt|xt−1, yt) :
– expansions of the p.d.f. or log(p.d.f.), Taylor
– mixture models : Gaussian i πiN(µi, σ2
i )
– Accept/Reject schemes
– Markov chain schemes : Metropolis-Hastings, Gibbs sampler
– dynamic stochastic simulation (Hybrid Monte Carlo)
– augmented sampling spaces :
→ slice samplers
→ auxiliary variables
23

Auxiliary variables
Pitt and Shephard 99 : approximating predictive p(xt|x
(k)
t−1, yt)
via augmented sampling space → p(xt, k|x
(k)
t−1, yt)
x_t
p(x_t/y_t)
x_t^(j)
x_t−1^(1)0
x_t−1
p(x_t−1/y_t−1)
1 x_t−1^(j) 3 x_t−1^(k)
x_t−1(N)0
x_t−1^(i)2
x_t^(i2)x_t^(i1) x_t^(k1) x_t^(k3)
x_t^(k2)
index of particle k (→ number of oﬀsping(s) of particle x
(k)
t−1) ∼ .|yt
⇒ boost particles with high likelihood
24

sampling/approximating predictive πt(xt|x0:t−1) may not be sufficient
for diffusing particles efficiently : e.g. discrepancy (πt)t>0 high :
⇒ consider a block of variables xt−L:t for a fixed lag L
27

Approaches using a block of variables
– discrete distributions : Meirovitch 85
– reweighting before resampling :
auxiliary variables Pitt and Shephard 99,
Wang et al. 02
⇒ discrete distribution → analytical form for proposal
xt ∼ πt+L(xt|x0:t−1) = πt+L(xt:t+L|x0:t−1)dxt+1:t+L
Meirovitch 85 : growing a polymer, random walk in discrete space
→ complexity X L
for lag L
28

Reweighting + resampling
2
1
01
0
0
0
0
1 1
31

auxiliary variables → resampling from (x
(j)
t )1≤j≤R :
→ propagate/sample x
(j)
t+1 → x
(j)
t+L with prior transitions p(xt|xt−1)
→ use second stage weights : w
(j)
t ∝ w
(j)
t−1 × iw
iw ∝
p(yt+L|x
(j)
t+L) . . . p(yt|x
(j)
t )
p(yt+L|µ
(kj )
t+L) . . . p(yt|µ
(kj )
t )
for resampling N paths (x
(i)
0:t)1≤i≤N
33

→ reweighting before resampling : Wang et al. 02
0
x_t^(i)
x_t
t
w_t^(i)
a_t^(i)
t t+L
x_t+L^(i_j)
a_t^(i_j)
propagate particles x
(i)
t → x
(ij )
t+1:t+L for j = 1, . . . , R
compute weights a
(ij )
t , particle path x
(i)
0:t reweighted with e.g.
a
(i)
t =
R
j=1 a
(ij )
t
α
, resampling from the set (x
(i)
0:t, a
(i)
t )i=1,...,N
34

Reweighting
→ need to sample/propagate xt from a/by block of variables :
πt+L(xt|x0:t−1) = πt+L(xt:t+L|x0:t−1)dxt+1:t+L
⇒ sampling a block of variables
→ design a proposal/candidate distribution
35

Sampling recursively a block of variables
t−L t−L+1 tt−1
xt−L:t−1 → xt−L+1:t : imputing xt and re-imputing xt−L+1:t−1
36

Sampling a block of variables
t−L+1x’(
0
:0 t−1x( )
:t)
direct sampling :
xt−L+1:t ∼ qt(xt−L+1:t|x0:t−1)
37

t−L+1x’(
t−L+1x(
0
:0 t−1x(
:0 t−Lx(
:t)
)
)
)t−1:
proposal/candidate distribution for the block :
(x0:t−L, xt−L+1:t) ∼ πt−1(x0:t−1)qt(xt−L+1:t|x0:t−1)dxt−L+1:t−1
38

⇒ Idea : consider extended block of variables
(x0:t−L, xt−L+1:t) → (x0:t−L, xt−L+1:t−1, xt−L+1:t) = (x0:t−1, xt−L+1:t)
t−L+1x’(
t−L+1x(
0
:0 t−Lx(
:t)
)
)t−1:
39

candidate distribution for extended block
(x0:t−L, xt−L+1:t−1, xt−L+1:t) = (x0:t−1, xt−L+1:t) :
(x0:t−1, xt−L+1:t) ∼ πt−1(x0:t−1)qt(xt−L+1:t|x0:t−1)
direct sampling :
(x0:t−L, xt−L+1:t) ∼ πt−1(x0:t−1)qt(xt−L+1:t|x0:t−1)dxt−L+1:t−1
40

target distribution for the block (x0:t−L, xt−L+1:t) :
πt(x0:t−L, xt−L+1:t)
⇒ auxiliary target distribution for the extended block
(x0:t−1, xt−L+1:t) = (x0:t−L, xt−L+1:t−1, xt−L+1:t) :
πt(x0:t−L, xt−L+1:t)rt(xt−L+1:t−1|x0:t−L, xt−L+1:t)
with rt = any conditional distribution
⇒ proposal + target distributions → importance sampling
41

Sequential Importance Block Sampling/Resampling
Simulation scheme t − 1 → t (index (i) dropped) :
– Proposal sampling step
xt−L+1:t ∼ qt(xt−L+1:t|x0:t−1)
– Updating weights
wt ∝ wt−1 ×
πt−1(x0:t−1)qt(xt−L+1:t|x0:t−1)
– Resampling step
42

Sampling techniques for a block of variables
sampling the block xt−L+1:t ∼ qt(xt−L+1:t|x0:t−1) :
→ forward-backward recursion : e. g. Carter and Kohn 94
43

Sampling techniques for a block of variables
→ forward-backward recursion :
→ approximations : expansions, mixture models, MCMC, . . .
44

Optimal proposal qt(xt−L+1:t|x0:t−1) distribution :
→ mimimizing variance of incremental weight w
(i)
t ∝ w
(i)
t−1 × iw :
iw =
πt−1(x0:t−1)qt(xt−L+1:t|x0:t−1)
⇒ qt = L-step ahead predictive
πt(xt−L+1:t|x0:t−L) = p(xt−L+1:t|xt−L, yt−L+1:t)
For one variable : optimal qt = 1-step ahead predictive
πt(xt|x0:t−1) = p(xt|xt−1, yt)
45

→ block of variables ⇒ optimal proposal and target distribution
mimimizing variance of incremental weight w
(i)
t ∝ w
(i)
t−1 × iw
iw =
πt−1(x0:t−1)qt(xt−L+1:t|x0:t−1)
→ optimal conditional distribution rt(xt−L+1:t−1|x0:t−L, xt−L+1:t)
⇒ rt = (L − 1)-step ahead predictive
πt−1(xt−L+1:t−1|x0:t−L) = p(xt−L+1:t−1|xt−L, yt−L+1:t−1)
46

Approximations for block sampling
Sampling (sub-)optimal qt(xt−L+1:t|x0:t−1) :
→ exact/approximated forward-backward recursions
→ approximations : expansions, mixture models, MCMC, . . .
For approximated optimal qt and rt, incremental weight :
πt(x0:t−L, xt−L+1:t)qt−1(xt−L+1:t−1|x0:t−L)
πt−1(x0:t−1)qt(xt−L+1:t|x0:t−L)
p(x0:t−L, xt−L+1:t|y1:t)q(xt−L+1:t−1|xt−L, yt−L+1:t−1)
p(x0:t−1|y1:t−1)q(xt−L+1:t|xt−L, yt−L+1:t)
48

Overview - Break
– Strategies for sampling :
→ “optimal” candidate distribution
sampling with e.g. auxiliary variables
→ considering a block of variables : reweighting
⇒ sampling a block of variables :
deﬁnition of importance sampling for a block
performing sampling → “optimal” candidate distribution
– ⇒ Examples, applications :
→ simple, complex models
→ why the sampling strategy for particles can be crucial ?
49

Example
Linear and Gaussian state space model :
xt = αxt−1 + ut x0, ut ∼ N(0, 1)
yt = xt + vt vt ∼ N(0, σ2
)
Sequential Monte Carlo methods :
– Bootstrap ﬁlter, proposal p(xt|xt−1)
– SISR with optimal proposal p(xt|xt−1, yt)
– SISR for blocks with optimal proposal p(xt−L+1:t|xt−L, yt−L+1:t)
computed by forward-backward exact recursions
⇒ estimates compared with Kalman ﬁlter results
⇒ approximation of target distribution p(xt|y1:t)
50

Estimation
0 10 20 30 40 50 60 70 80 90 100
−4
−3
−2
−1
0
1
2
3
4
time index
x(t)
x(t) and estimates
model (α, σ) = (0.9, 0.1) xt =
N
i=1 w
(i)
t x
(i)
t N=100
51

Approximation of the target distribution
⇒ Eﬀective Sample Size :
ESS =
1
N
i=1[w
(i)
t ]2
w(i)
= 1
N : ESS = N
pi(x_t)
x_t
w(i)
≈ 0 ∀i except one : ESS = 1
x_t
pi(x_t)
⇒ Resampling performed for ESS ≤ N
2 , N
10
52

Resampling for ESS ≤ N
2 , N = 100
0 10 20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
60
70
80
90
100
time index
ESS
Efficient Sample Size (ESS)
ESS for Bootstrap (→), SISR (→) and SISR for blocks
of 2 variables (→) with optimal proposals
53

2 , N = 100, 100 time steps
algorithm ESS resampling steps CPU time
Bootstrap 11.19 99 0.84
optimal SISR 77.1 2 0.12
Block-SISR L = 2 99.1 1 0.23
54

2 , N = 100, ∞ time steps
algorithm ESS resampling steps CPU time
Bootstrap 10 100% ∝0.84
optimal SISR 75 0.04% ∝0.12
Block-SISR L = 2 99 0% ∝0.23
55

2 , various N
algorithm ESS resampling steps
Bootstrap 10%N 100%
optimal SISR 75%N 0.04%
Block-SISR L = 2 99%N 0%
computational complexity : resampling O(N) → CPU time
56

CPU time / number of particles N
2 , 1,000 time steps
100 150 200 250 300 350 400 450 500
0
5
10
15
20
25
30
35
40
Number of particles N
CPUtime
Bootstrap (→), SISR (→) and SISR for blocks
57

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
0
10
20
30
40
50
60
70
number of particles N
CPUtime
CPU time/number of particles N
SISR (→) and SISR for blocks
58

for this model :
– Same estimation results as Kalman ﬁltering
– sampling ⇒ = approximation for the target distribution
– N ≤ 500 : computational complexity, CPU time
→ SISR with optimal proposal p(xt|xt−1, yt)
– N ≥ 500 : → block SISR with optimal proposal
p(xt−L+1:t|xt−L, yt−L+1:t)
59

Sampling strategies
xt = αxt−1 + ut x0, ut ∼ N(0, σ2
u)
v)
– σv=0.1 → observation yt very informative/prior σu=1.0
⇒ take into account yt for diﬀusing particles
p(xt|xt−1) → p(xt|xt−1, yt) ⇒ ESS
– α=0.9 → variables (xt)t correlated
⇒ sampling by block xt−L+1:t
block of observations yt−L+1:t more informative/single one yt
p(xt|xt−1, yt) → p(xt−L+1:t|xt−L, yt−L+1:t) ⇒ ESS
60

2 , N = 100
0 20 40 60 80 100
0
20
40
60
80
100
0 20 40 60 80 100
0
20
40
60
80
100
120
0 20 40 60 80 100
0
20
40
60
80
100
120
0 20 40 60 80 100
0
20
40
60
80
100
ESS for Bootstrap (→), SISR with optimal proposals for 1 (→),
2 (→) and 10 variables (→)
σu=1.0, left/right : σv=0.1/1.0, top/bottom : α=0.9/0.5
61

100 200 300 400 500
0
10
20
30
40
100 200 300 400 500
0
5
10
15
20
100 200 300 400 500
0
5
10
15
20
100 200 300 400 500
0
10
20
30
40
Bootstrap (→), SISR with optimal proposals for 1 (→),
62

0 2000 4000 6000 8000 10000
0
200
400
600
800
0 2000 4000 6000 8000 10000
0
100
200
300
400
0 2000 4000 6000 8000 10000
0
200
400
600
800
0 2000 4000 6000 8000 10000
0
50
100
150
200
250
300
350
SISR with optimal proposals for 1 (→), 2 (→) and 10 variables (→)
63

2 , N = 100
0 20 40 60 80 100
0
20
40
60
80
100
0 20 40 60 80 100
30
40
50
60
70
80
90
100
0 20 40 60 80 100
0
20
40
60
80
100
0 20 40 60 80 100
40
50
60
70
80
90
100
ESS for Bootstrap (→), SISR with optimal proposals for 1 (→),
64

100 200 300 400 500
0
5
10
15
20
100 200 300 400 500
0
5
10
15
20
25
100 200 300 400 500
0
5
10
15
20
100 200 300 400 500
0
5
10
15
20
Bootstrap (→), SISR with optimal proposals for 1 (→),
65

0 2000 4000 6000 8000 10000
0
100
200
300
400
0 2000 4000 6000 8000 10000
0
50
100
150
200
250
0 2000 4000 6000 8000 10000
0
50
100
150
200
250
300
350
0 2000 4000 6000 8000 10000
0
50
100
150
200
SISR with optimal proposals for 1 (→), 2 (→) and 10 variables (→)
66

Overview- Break
– Strategies for sampling
– Applications : Linear and Gaussian model
⇒ sampling strategy :
→ approximation of the target distribution, CPU time
→ information in observation, dynamic of the state variable
→ nonlinear non-Gaussian models
67

Example
Nonlinear state space model :
xt = α(xt−1 + βx3
t−1) + ut x0, ut ∼ N(0, σ2
u)
v)
Sequential Monte Carlo methods :
– Bootstrap ﬁlter, proposal p(xt|xt−1)
– SISR with optimal proposal p(xt|xt−1, yt) approximated by
KF/EKF
– SISR for blocks with optimal proposal p(xt−L+1:t|xt−L, yt−L+1:t)
approximated by forward-backward recursions with KF/EKF
Parameters values α=0.9, β=0.2, σu=0.1 and σv=0.05
⇒ approximation of target distribution p(xt|y1:t)
68

Simulation results
algorithm MSE ESS RS CPU
Bootstrap 0.0021 36.8 70.3 % 0.68
SISR-KF 0.0019 64.7 19.3% 0.44
SISR-EKF 0.0019 65.8 19.2% 0.48
BSISR-KF 0.0018 72.3 0.9% 0.21
BSISR-EKF 0.0018 73.5 0.8% 0.24
N = 100 particles, 100 runs of particle ﬁlters for a single and for a
block of L = 2 variables (MSE from KF/EKF = 0.0034).
69

2 , N = 100
0 20 40 60 80 100 120 140 160 180 200
0
10
20
30
40
50
60
70
80
90
100
time index
EffectiveSampleSize
Approximated ESS vs. time index for a realization of the Bootstrap
ﬁlter (dotted), the SISR with Kalman ﬁlter proposal for a single
variable (dashdotted) and for a block of L=2 variables (straight).
70

Simulation results
block size L N=100 N=500 N=1000 RS
2 74 370 715 0.9%
3 96 493 985 0.9%
4 99 496 989 1%
5 98 494 988 1%
10 97 486 972 2.5%
Approximated ESS averaged over 100 runs of particle ﬁlters for
blocks of L variables, considering N particles.
71

100 200 300 400 500 600 700 800 900 1000
0
1
2
3
4
5
6
7
Number of Particles
ComputingTime
CPU time vs. N for bootstrap ﬁlter (dotted), SISR with KF proposal
for a single variable (KF : dashed, EKF : dashdotted) and for a block
of L=2 variables (straight), 100 realizations.
72

2000 4000 6000 8000 10000
0
2
4
6
8
10
12
14
Number of Particles
ComputingTime
Computational time vs. N for block sampling scheme with lags from
L=2 (bottom), 3, 4, 5, 10 (top), 100 realizations.
73

for this model :
– Good approximations of the target distribution
– sampling ⇒ = approximation for the target distribution
– even for small N, block SISR with approximated optimal proposal
p(xt−L+1:t|xt−L, yt−L+1:t) is eﬃcient for L=3, 4, 5
– → information in observation : σu=0.1, σv=0.05
74

Conclusion
⇒ Importance of proposal/candidate distribution for
Sequential Monte Carlo simulation methods
design of proposal :
→ information in observation, dynamic of the state variable :
p(xt|xt−1) ←→ p(xt|yt, xt−1) ←→ p(xt|yt)
→ sampling a block/ﬁxed lag of variables can be useful :
– for intermittent/informative observation, correlated variables
– applications ⇒ tracking, radar, navigation, positioning . . .
75

References - SISR, Sequential Monte Carlo
– N. Gordon, D. Salmond, and A. F. M. Smith, “Novel approach to
nonlinear and non-Gaussian Bayesian state estimation,”
Proceedings IEE-F, vol. 140, pp. 107–113, 1993.
– G. Kitagawa, “Monte carlo ﬁlter and smoother for non-Gaussian
nonlinear state space models,” J. Comput. Graph. Statist., vol. 5,
pp. 1–25, 1996.
– A. Doucet, N. de Freitas, and N. Gordon, Eds., Sequential Monte
Carlo methods in practice, Statistics for engineering and
information science. Springer, 2001.
76

References - block/fixed lag approaches
– H. Meirovitch, “Scanning method as an unbiased simulation
technique and its application to the study of self-avoiding random
walks,” Phys. Rev. A, vol. 32, pp. 3699–3708, 1985.
– M. K. Pitt and N. Shephard, “Filtering via simulation : auxiliary
particle filter,” J. Am. Stat. Assoc., vol. 94, pp. 590–599, 1999.
– X. Wang, R. Chen, and D. Guo, “Delayed-pilot sampling for
mixture Kalman filter with application in fading channels,” IEEE
Trans. Sig. Proc., vol. 50, pp. 241–253, 2002.
77

References - block/fixed lag sampling methods
– A. Doucet and S. Sénécal, “Fixed-Lag Sequential Monte Carlo”,
accepted at EUSIPCO 2004.
– S. Sénécal and A. Doucet, “An example of sequential Monte Carlo
block sampling method,” AIC2003 Science of Modeling,
pp. 418-419, 2003.
– C. K. Carter and R. Kohn, “On the Gibbs sampling for state space
models,” Biometrika, vol. 81, pp. 541–553, 1994.
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Sampling strategies for Sequential Monte Carlo (SMC) methods

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