-arnaud doucet, nando de freitas et al, uai 2000-
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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-
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outlineoutline
IntroductionProblem FormulationImportance Sampling and Rao-BlackwellisationRao-Blackwellisation Particle FilterExampleConclusion
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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.
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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
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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
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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
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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
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3.Importance Sampling and Rao3.Importance Sampling and Rao-Blackwellisation-Blackwellisation
Monte Carlo integration
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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
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Point mass approximation
Normalized
Importance weight
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In case, we can marginalize out analytically
tx :0
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ExampleExample
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We can estimate with a reduced variance)( tfI
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4.Rao-Blackwellisation Particle Filters4.Rao-Blackwellisation Particle Filters
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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
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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
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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)
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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
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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..