The Hierarchical Particle Filter
Adam M. Johansen and Arnaud Doucet
http://go.warwick.ac.uk/amjohansen/talks
MCMSki V Lenzerheide
7th January 2016
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
Context & Outline
Filtering in State-Space Models:
I
SIR Particle Filters [GSS93]
I
Block-Sampling Particle Filters [DBS06]
Exact Approximation of Monte Carlo Algorithms:
I
Particle MCMC [ADH10] and SMC2 [CJP13]
Exact Approximation and Particle Filters:
I
Approximated RBPFs [CSOL11] Exactly [JWD12]
I
Hierarchical SMC [Here]
I
Pseudomarginal State Augmentation: More of the SAME?
[Axel Finke; this session]
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
Hidden Markov Models
Consider:
x1
x2
x3
x4
x5
x6
y1
y2
y3
y4
y5
y6
I
Unobserved Markov chain {Xn } transition f .
I
Observed process {Yn } conditional density g.
Density:
I
p(x1:n , y1:n ) = f1 (x1 )g(y1 |x1 )
n
Y
f (xi |xi−1 )g(yi |xi ).
i=2
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
Sequential Importance Resampling
I
Algorithmically, at iteration n:
I
I
I
i
i
, X1:n−1
} for i = 1, . . . , N :
Given {Wn−1
ei
Resample, obtaining {1/N, X
1:n−1 }.
I
i
en−1
Sample Xni ∼ qn (·|X
)
I
Weight Wni ∝
i ei
i
f (Xn
|Xn−1 )g(yn |Xn
)
ei
qn (X i |X
)
n
n−1
Actually:
I
I
Resample efficiently.
Only resample when necessary.
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
Particle MCMC
I
I
MCMC algorithms which employ SMC proposals [ADH10]
SMC algorithm as a collection of RVs
I
I
I
I
Construct MCMC algorithms:
I
I
I
I
I
Values
Weights
Ancestral Lines
With many auxiliary variables
Exactly invariant for distribution on extended space
Standard MCMC arguments justify strategy
SMC2 employs the same approach within an SMC setting.
What else does this allow us to do with SMC?
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
Ancestral Trees
t=1
t=2
t=3
a13 =1
a43 =3
a12 =1
a42 =3
b23,1:3 =(1, 1, 2) b43,1:3 =(3, 3, 4) b63,1:3 =(4, 5, 6)
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
SMC Distributions
Formally gives rise to the SMC Distribution:
M
an−L+2:n , xn−L+1:n , k; xn−L
ψn,L
#
"
"M
#
n
M
Y
Y
Y
aip
i
i
r(ap |wp−1 )
q xp |xp−1
r(k|wn )
=
q( xn−L+1 xn−L )
i=1
p=n−L+2
i=1
and the conditional SMC Distribution:
ek
k
M
k
ek
e
ψen,L
a
,
x
;
x
b
,
k,
x
e
n−L
n−L+1:n−1
n−L+1:n
n−L+2:n
n−L+1:n
M
en−L+1:n , k; xn−L )
ψn,L
(e
an−L+2:n , x
"
= #
ebk
n
e
e
Q
bk
bn
n,n−L+1
n,p
n,p−1
k
e
e p−1 q x
e n)
q x
en−L+1 |xn−L
r bn,p |w
ep |e
xp−1
r(k|w
p=n−L+2
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
Block Sampling: An Idealised Approach
At time n, given x1:n−1 ; discard xn−L+1:n−1 :
I
Sample from q(xn−L+1:n |xn−L , yn−L+1:n ).
I
Weight with
W (x1:n ) =
I
p(x1:n |y1:n )
p(x1:n−L |y1:n−1 )q(xn−L+1:n |xn−L , y1:n−L+1:n )
Optimally,
q(xn−L+1:n |xn−L , yn−L+1:n ) =p(xn−L+1:n |xn−L , yn−L+1:n )
p(x1:n−L |y1:n )
W (x1:n ) ∝
=p(yn |x1:n−L , yn−L+1:n−1 )
p(x1:n−L |y1:n−1 )
I
Typically intractable; auxiliary variable approach in [DBS06].
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
Local Particle Filtering: Current Trajectories
Starting Point
4
2
0
−2
−4
0
2
The Hierarchical Particle Filter
Adam M. Johansen
4
6
8
n
10
12
14
16
Introduction
Block Sampling Particle Filters
References
Local Particle Filtering: PF Proposal
PF Step
4
2
0
−2
−4
0
2
The Hierarchical Particle Filter
Adam M. Johansen
4
6
8
n
10
12
14
16
Introduction
Block Sampling Particle Filters
References
Local Particle Filtering: CPF Auxiliary Proposal
CPF Step
4
2
0
−2
−4
0
2
The Hierarchical Particle Filter
Adam M. Johansen
4
6
8
n
10
12
14
16
Introduction
Block Sampling Particle Filters
References
Local SMC: Version 1
I
I
Not just a Random Weight Particle Filter.
Propose from:
⊗n−1
M
U1:M
(b1:n−2 , k)p(x1:n−1 |y1:n−1 )ψn,L
(an−L+2:n , xn−L+1:n , k; xn−L )
M
ek
ek
ψen−1,L−1
a
n−L+2:n−1 , x
n−L+1:n−1 ; xn−L ||bn−L+2:n−1 , xn−L+1:n−1
I
Target:
k̄
b
n,n−L+1:n
⊗n
U1:M
(b1:n−L , b̄k̄n,n−L+1:n−1 , k̄)p(x1:n−L , xn−L+1:n
|y1:n )
k̄
k̄
bn,n−L+1:n
k
k
M
e
ψn,L an−L+2:n , xn−L+1:n ; xn−L b̄n,n−L+1:n , xn−L+1:n
M
en−L+1:n−1 , k; xn−L ) .
ψn−1,L−1
(e
an−L+2:n−1 , x
I
Weight: Z̄n−L+1:n /Zen−L+1:n−1 .
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
Stochastic Volatility Bootstrap Local SMC
I
Model:
f (xi |xi−1 ) =N φxi−1 , σ 2
g(yi |xi ) =N 0, β 2 exp(xi )
I
Top Level:
I
I
I
Local SMC proposal.
Stratified resampling when ESS< N/2.
Local SMC Proposal:
I
Proposal:
q(xt |xt−1 , yt ) = f (xt |xt−1 )
I
Weighting:
W (xt−1 , xt ) ∝
I
f (xt |xt−1 )g(yt |xt )
= g(yt |xt )
f (xt |xt−1 )
Resample residually every iteration.
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
SV Exchange Rata Data
Exchange Rate Data
5
4
Observed Values
3
2
1
0
−1
−2
−3
−4
0
100
200
300
400
500
600
700
800
900
n
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
SV Bootstrap Local SMC: M=100
N = 100, M = 100
100
Average Number of Unique Values
90
80
70
L=2
L=4
L=6
L = 10
60
50
40
30
20
10
0
0
100
200
300
400
500
600
700
800
900
n
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
SV Bootstrap Local SMC: M=1000
N = 100, M = 1000
100
Average Number of Unique Values
90
80
70
60
L=2
L=4
L=6
L = 10
L = 14
L = 18
50
40
30
20
10
0
0
100
200
300
400
500
600
700
800
900
n
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
SV Bootstrap Local SMC: M=10000
N=100, M=10,000
100
Average Number of Unique Values
90
80
70
60
50
L=2
L=4
L=6
L = 10
L = 14
L = 18
L = 22
L = 26
40
30
20
10
0
0
100
200
300
400
500
600
700
800
900
n
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
Local SMC: Version 2
Problems with this PF+CPF scheme:
I Expensive to run 2 filters per proposal. . .
I and large M is required. . .
I can’t we do better?
Using a non-standard CPF/PF proposal is preferable:
I Running a CPF from time n − L + 1 to n,
I and extending it’s paths with a PF step to n + 1,
I invoking a careful auxiliary variable construction,
I we reduce computational cost and variance.
I Leads to importance weights:
bk
∝
n−1,n−L
p̂(yn−L+1:n |xn−L
)
bk
n−1,n−L
p̂(yn−L+1:n−1 |xn−L
)
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
Hierarchical Particle Filtering: Current Trajectories
Starting Point
7
6
5
4
3
2
1
0
-1
-2
0
2
4
6
8
10
12
14
16
n
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
Hierarchical Particle Filtering: CPF Proposal Component
CPF Step
7
6
5
4
3
2
1
0
-1
-2
0
2
4
6
8
10
12
14
16
n
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
Hierarchical Particle Filtering: PF Proposal Component
PF Step
7
6
5
4
3
2
1
0
-1
-2
0
2
4
6
8
10
12
14
16
n
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
A Toy Model: Linear Gaussian HMM
I
Linear, Gaussian state transition:
f (xt |xt−1 ) = N (xt ; xt−1 , 1)
I
and likelihood
g(yt |xt ) = N (yt ; xt , 1)
I
Analytically: Kalman filter/smoother/etc.
I
Simple bootstrap HPF:
I
Local proposal:
q(xt |xt−1 , yt ) = f (xt |xt−1 )
I
Weighting:
W (xt−1 , xt ) ∝ g(yt |xt )
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
●
3
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
N
●
●
●
●
●
100
●
●
●
●
●
1000
●
10000
●
●
1e+05
2
●
●
M
●
●
1
●
rZ
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
● ●
●
●
1
●
● ●
● ●
●
2
●
●
●
4
8
16
32
64
●
●
128
●
●
256
●
●
●
●
●
512
●
1024
●
Bo
o
tst
4
rap
0
L
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
2
N
●
100
1000
10000
●
●
●
●
●
●
●
●
1e+05
●
0
●
●
●
●
M
1
●
2
4
log(rZ)
●
8
●
16
32
−2
64
128
256
512
1024
L
−4
4
00
10
00
0
12
80
0
16
00
0
25
60
0
32
00
0
51
20
0
64
00
0
1e
+0
5
10
24
00
12
80
00
20
48
00
25
60
00
40
96
00
51
20
00
10
24
00
0
20
48
00
0
40
96
00
0
00
80
0
00
64
32
16
80
00
Bootstrap
C
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
In Conclusion
I
SMC can be used hierarchically.
I
Software implementation is not difficult
[Joh09, Zho13, Mur13, TCF+ 14].
I
The optimal block-sampling particle filter can be
approximated exactly
I
but at a considerable computational cost.
I
Many related things are also possible. . .
I
I
I
EA-RBPFs [JWD12]
Tempering in blocks [Joh15]
Iterated Particle Filters [GJL15]
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
References I
[ADH10] C. Andrieu, A. Doucet, and R. Holenstein. Particle Markov chain Monte Carlo. Journal of the Royal
Statistical Society B, 72(3):269–342, 2010.
[CJP13] N. Chopin, P. Jacob, and O. Papaspiliopoulos. SMC2 : an efficient algorithm for sequential analysis of
state space models. Journal of the Royal Statistical Society B, 75(3):397–426, 2013.
[CSOL11] T. Chen, T. Schön, H. Ohlsson, and L. Ljung. Decentralized particle filter with arbitrary state
decomposition. IEEE Transactions on Signal Processing, 59(2):465–478, February 2011.
[DBS06] A. Doucet, M. Briers, and S. Sénécal. Efficient block sampling strategies for sequential Monte Carlo
methods. Journal of Computational and Graphical Statistics, 15(3):693–711, 2006.
[GJL15] P. Guarniero, A. M. Johansen, and A. Lee. The iterated auxiliary particle filter. ArXiv mathematics
e-print 1511.06286, ArXiv Mathematics e-prints, 2015.
[GSS93] N. J. Gordon, S. J. Salmond, and A. F. M. Smith. Novel approach to nonlinear/non-Gaussian Bayesian
state estimation. IEE Proceedings-F, 140(2):107–113, April 1993.
[Joh09] A. M. Johansen. SMCTC: Sequential Monte Carlo in C++. Journal of Statistical Software,
30(6):1–41, April 2009.
[Joh15] A. M. Johansen. On blocks, tempering and particle MCMC for systems identification. In Proceedings
of 17th IFAC Symposium on System Identification, Beijing, China., 2015. IFAC. To appear. Invited
submission.
[JWD12] A. M. Johansen, N. Whiteley, and A. Doucet. Exact approximation of Rao-Blackwellised particle
filters. In Proceedings of 16th IFAC Symposium on Systems Identification, pages 488–493, Brussels,
Belgium, July 2012. IFAC, IFAC.
[Mur13] L. M. Murray. Bayesian state-space modelling on high-performance hardware using LibBi.
Mathematics e-print 1306.3277, ArXiv, 2013.
[TCF+ 14] A. Todeschini, F. Caron, M. Fuentes, P. Legrand, and P. Del Moral. Biips: Software for Bayesian
inference with interacting particle systems. Mathematics e-print, ArXiv, 2014.
[Zho13] Y. Zhou. vSMC: Parallel sequential Monte Carlo in C++. Mathematics e-print 1306.5583, ArXiv,
2013.
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References
Key Identity
M
ψn,L
(an−L+2:n , xn−L+1:n , k; xn−L )
k
M (ak
p(xn−L+1:n |xn−L , yn−L+1:n )ψen,L
n−L+2:n , xn−L+1:n , k; xn−L ||. . . )
"
k
k
#
n
Q
bn,n−L+1
bn,p bn
n,p−1
k
q xn−L+1 |xn−L
r bn,p |wp−1 q xp |xp−1
r(k|wn )
=
p=n−L+2
p(xn−L+1:n |xn−L , yn−L+1:n )
b
=Zn−L+1:n /p(yn−L+1:n |xn−L )
The Hierarchical Particle Filter
Adam M. Johansen
Introduction
Block Sampling Particle Filters
References