Block-Tempered Particle Filters
Adam M. Johansen
University of Warwick
a.m.johansen@warwick.ac.uk
www2.warwick.ac.uk/fac/sci/statistics/staff/academic/johansen/talks/
SysID 17: October 20th, 2015
Outline
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Background: SMC and PMCMC
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Tempering Blocks in SMC
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Example: proof-of-concept results
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Conclusions
Filtering
Online inference for State Space Models:
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x1
x2
x3
x4
x5
x6
y1
y2
y3
y4
y5
y6
Given transition fθ (xn |xn−1 ),
and likelihood gθ (yn |xn ),
use pθ (xn |y1:n ) to characterize latent state, but,
R
pθ (xn−1 |y1:n−1 )fθ (xn |xn−1 )dxn−1 gθ (yn |xn )
pθ (xn |y1:n ) = R R
pθ (xn−1 |y1:n−1 )fθ (x0n |xn−1 )dxn−1 gθ (yn |x0n )dx0n
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isn’t often tractable.
Particle Filtering
A (sequential) Monte Carlo (SMC) scheme to approximate the
filtering distributions.
A Simple Particle Filter
At n = 1:
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Sample X11 , . . . , X1N ∼ µθ .
For n > 1:
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Sample
j
j
j=1 gθ (yn−1 |Xn−1 )fθ (·|Xn−1 )
Pn
k
k=1 gθ (yn−1 |Xn−1 )
PN
Xn1 , . . . , XnN
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∼
Approximate pθ (dxn |y1:n ), pθ (y1:n ) with
PN
pbθ (·|y1:n ) =
j=1
P
n
gθ (yn |Xnj )δXnj
k
k=1 gθ (yn |Xn )
N
,
pbθ (y1:n )
1X
gθ (yn |Xnk )
=
pbθ (y1:n−1 )
n j=1
Online Particle Filters for Offline Systems Identification
Particle Markov chain Monte Carlo (PMCMC) [ADH10]
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Embed SMC within MCMC,
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justified via explicit auxiliary variable construction,
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and in some simple cases by a pseudomarginal [AR09]
argument.
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Very widely applicable,
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but prone to poor mixing when SMC performs poorly for
some θ [OWG15, Section 4.2.1].
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Is valid for very general SMC algorithms.
Block-Sampling and Tempering in Particle Filters
Tempered Transitions [GC01]
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Introduce each likelihood term gradually, targetting:
πn,m (x1:n ) ∝ p(x1:n |y1:n−1 )p(yn |xn )βm
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between p(x1:n−1 |y1:n−1 ) and p(x1:n |y1:n ).
Can improve performance — but to a limited extent.
Block Sampling [DBS06]
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Essentially uses yn:n+L in proposing xn .
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Can dramatically improve performance,
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but requires good analytic approximation of
p(xn:n+L |xn−1 , yn:n+L ).
Block-Tempering
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We could combine blocking and tempering strategies.
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Run a simple SMC sampler [DDJ06] targetting:
1
β(t,r)
θ
πt,r
(x1:t∧T ) =µθ (x1 )
T
∧t
Y
s=2
1
γ(t,r)
gθ (y1 |x1 )
fθ (xs |xs−1 )
s
β(t,r)
·
s
γ(t,r)
gθ (ys |xs )
s
s
where {β(t,r)
} and {γ(t,r)
} are [0, 1]-valued
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,
for s ∈ [[1, T ]], r ∈ [[1, R]] and t ∈ [[1, T 0 ]], with T 0 = T + L
for some R, L ∈ N.
Can be validly embedded within PMCMC:
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Terminal likelihood estimate is unbiased.
Explicit auxiliary variable construction is possible.
(1)
Two Simple Block-Tempering Strategies
Tempering both likelihood and transition probabilities
s
β(t,r)
=
s
γ(t,r)
R(t − s) + r
= 1∧
∨0
RL
(2)
Tempering only the observation density
s
β(t,r)
=I{s ≤ t}
s
γ(t,r)
R(t − s) + r
= 1∧
∨ 0,
RL
(3)
Tempering Only the Observation Density
s
s
β(t,R)
= β(t,0)
1
0
t
s
s
γ(t,R)
1
s
γ(t,0)
1
L
0
t−L+1
t
s
Illustrative Example
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Univariate Linear Gaussian SSM:
transition f (x0 |x) =N (x0 ; x, 1)
likelihood
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g(y|x) =N (y; x, 1)
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Artificial jump in observation
seqeuence at time 75.
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Cartoon of model misspecification
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— a key difficulty with PMCMC.
Observations, y[t]
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Temper only likelihood.
−15
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Use single-site Metropolis-Within
Gibbs (standard normal
proposal) MCMC moves.
0
25
50
Time, t
75
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100
True Filtering and Smoothing Distributions
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Algorithm
smoother
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25
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Time, t
75
100
Estimating the Normalizing Constant, Zb
−88
L
log(Estimated Z)
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−90
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1
2
3
5
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R
● 0
1
5
10
−92
3
4
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log(Cost = N(1+RL))
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Relative Error in Zb Against Computational Effort
2
L
log(Relative Error in Z)
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Performance:
log(N)
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R
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0
0
1
5
10
4
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log(Cost = N(1+RL))
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(R = 1, L = 1)
≺(R > 1, L = 1)
≺(R > 1, L > 1)
Conclusions
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PMCMC is perhaps even more powerful than has yet been
recognised.
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Exploiting the offline nature of the PMCMC setting allows
more flexibility than the filtering framework.
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The particular block-tempered approach developed here
warrants further investigation.
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Another approach to similar problems is provided by: the
iAPF [GJL15, preprint on ArXiv RSN]
Conclusions
I
PMCMC is perhaps even more powerful than has yet been
widely recognised.
I
Exploiting the offline nature of the PMCMC setting allows
more flexibility than the filtering framework.
I
The particular block-tempered approach developed here
warrants further investigation.
I
Another approach to similar problems is provided by: the
iAPF [GJL15, preprint on ArXiv RSN]
Thanks for listening. . .
Any Questions?
References
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Particle Markov chain Monte Carlo.
Journal of the Royal Statistical Society B, 72(3):269–342, 2010.
C. Andrieu and G. O. Roberts.
The pseudo-marginal approach for efficient Monte Carlo computations.
Annals of Statistics, 37(2):697–725, 2009.
A. Doucet, M. Briers, and S. Sénécal.
Efficient block sampling strategies for sequential Monte Carlo methods.
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P. Del Moral, A. Doucet, and A. Jasra.
Sequential Monte Carlo samplers.
Journal of the Royal Statistical Society B, 63(3):411–436, 2006.
S. Godsill and T. Clapp.
Improvement strategies for Monte Carlo particle filters.
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Methods in Practice, Statistics for Engineering and Information Science,
pages 139–158. Springer Verlag, New York, 2001.
P. Guarniero, A. M. Johansen, and A. Lee.
The iterated auxiliary particle filter.
In preparation, 2015.
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Likelihood free inference for markov processes: a comparison.
Statistical Applications in Genetics and Molecular Biology, 2015. In press.