1. Rao-Blackwellised Particle Filtering for Dynamic Bayesian Network Arnaud Doucet Nando de Freitas Kevin Murphy Stuart Russell

2. Introduction  Sampling in high dimension  Model has tractable substructure  Analytically marginalized out, conditional on certain other nodes being imputed  Using Kalman filter, HMM filter, junction tree algorithm for general DBNs  Reduce size of the space over we need to sample  Rao-Blackwellisation  marginalize out some of the variables

3. Problem Formulation  general state space model/DBN with hidden variables and observed variables.  is a Markov process of initial distribution  Transition equation  Observations  Estimate  recursion  not analytically, numerical approximation scheme

4. Cont’d  Divide hidden variables into two groups and  Conditional posterior distribution is analytically tractable.  Focus on estimating (reduced dimension)  Decomposition of posterior from chain rule  Marginal distribution

5. Importance sampling and RAO-Blackwellisation  Sample N i.i.d. random samples(particles) according to  Empirical estimate  Expected value of any function of hidden variables

6. Cont’d  Strong law of large numbers  converges almost surely towards as  Central limit theorem

7. Importance Sampling  impossible to sample efficiently from target  Importance distribution q  Easy to sample  p>0 implies q>0

8. Cont’d  The case where one can marginalize out analytically, propose alternative estimate for  Alternative importance sampling estimate of  To reach a given precision, will require a reduced number N of samples over

9. Rao-Blackwellised particle filters  Sequential importance sampling step  For i=1,…,N sample: and set:  For i=1,…,N evaluate importance weights up to a normalizing constant:  For i=1,…,N normalize importance weights:  Selection step  multiply/suppress samples with high/low importance weights to obtain random samples approximately distributed  MCMC step  Apply a markov transition kernel with invariant distribution given by to obtain to obtain

10. Robot localization and MAP building  Problem of concurrent localization and map learning  Location  Color of each grid cell  Observation  Basic idea of algorithm  Sample with PF  Marginalize out since they are conditionally independent given