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..