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Rao-Blackwellised Particle FilteriRao-Blackwellised Particle Filtering for Dynamic Bayesian Networng for Dynamic Bayesian Networ
ksks-Arnaud Doucet, Nando de Freitas
et al, UAI 2000-
outlineoutline
IntroductionProblem FormulationImportance Sampling and Rao-BlackwellisationRao-Blackwellisation Particle FilterExampleConclusion
IntroductionIntroductionFamous state estimaton algorithm, The Kalman filter and the HMM filter, are only applicable to linear-Gaussian models and if state space is so large, the computatuion cost becomes too expensive.Sequential Monte Carlo methods(Particle Filtering) have been introduced (Handschine and Mayne,1969) to handle large state model.
Particle Filtering(PF) = “condensation” = “sequential Monte Carlo” = “survival of the fittest”
PF can treat any type of probability distribution,nonlinearity and non-stationarity.
PF are powerful sampling based inference/learning algorithms for DBNs
Drawback of PF Inefficent in high-dimensional spaces (Variance becomes so large)
Solution Rao-Balckwellisation, that is, sample a subset of
the variables allowing the remainder to be integrated out exactly. The resulting estimates can be shown to have lower variance.
Rao-Blackwell Theorem
Problem FormulationProblem FormulationModel : general state space model/DBN with hidden variables and observed variables Objective:
or filtering density To solve this problem,one need approximation sc
hemes because of intractable integrals
tzty
)|( :1 tt yzp
Additive assumption in this paper: Divide hidden variables into two groups,
Conditional posterior distribution
is analytically tractable We only need to focus on estimating
Which lies in a space of reduced dimension
tt xandr
),|( :0:1:0 ttt ryxp)|( :1:0 tt yrp
3.Importance Sampling and Rao3.Importance Sampling and Rao-Blackwellisation-Blackwellisation
Monte Carlo integration
But it’s impossible to sample efficiently from the “target” posterior distribution .
Importance Sampling Method (Alternative way)
) | (:1 : 0 , : 0t t ty x r p
dxxgxg
xfdxxffI t )(
)(
)()()(
Weight function
Importance function
Point mass approximation
Normalized
Importance weight
In case, we can marginalize out analytically
tx :0
ExampleExample
We can estimate with a reduced variance)( tfI
4.Rao-Blackwellisation Particle Filters4.Rao-Blackwellisation Particle Filters
4.1Implementation Issues4.1Implementation IssuesSequential Importance Sampling Restrict importance function
We can obtain recursive formulas
and obtain “incremental weight” is given by
ttt wrwrw )()( 1:0:0
Choice of importance Distribution Simplest choice is to just sample from the
prior, => it can be inefficent, since it ignores the most recent evidence, .
“optimal” importance distribution:Minimizing the variance of the importance weig
ht.
)|( 1tt rrp
ty
But it is often too expensive.Several Deterministic approximations to the optimal distribution have been proposed, see for example(de Freitas 1999,Doucet 1998)Selection step Using Resampling : elimate samples with l
ow importance weight and multiply samples with high importance weight. ( ex: residual sampling, stratified sampling, multinomial sampling)
Examples: Examples: On-Line Regression and MoOn-Line Regression and Model Selection with Neural Networkdel Selection with Neural Network
Goal :
It is paossible to simulate and to compute coefficent analytically using Kalman filters.This is because the output of the neural network is linear in
ttt andku ,
t
t
Number of basis function
Conclusions and ExtensionsConclusions and Extensions
Successful application Conditionaliiy linear Gaussian state-space model
s Conditionally finite state-space HMMs
Possible extensions Dynamic models for counting observations Dynamic models with a time-varying unknown cov
ariance matrix for the dynamic noise Calsses of the exponential family state space mo
dels etc..
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