Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks Arnaud Doucet, Nando de Freitas, Kevin Murphy and Stuart Russell CS497EA presentation Hanning ZhouMarch 16, 2004

Sequential Monte Carlo for Dynamic Bayesian Networks Hanning Zhou CS497EA presentation March 16, 2004 Based on Arnaud Doucet, Nando de Freitas, Kevin Murphy and Stuart Russell’s tutorial & Arnaud Doucet, Simon Godsill and Christophe Andrieu’s review paper in Statistics and Computing (2000)

Introduction Bayesian filtering Given transition eqn. of a hidden Markov process x[t] obs. eqn. for likelihood of the observable y[t] y[0] ~ y[t] Recursively estimate x[t+1] Applications Target tracking (Gordon et al 1993) Blind de-convolution of comm. Channels (Liu and Chen 1995, Clapp and Godsill 1999) Speed/audio enhancement (Godsill and Rayne1998)

A sneak peek at particle filtering

Introduction (2) Analytical methods Kalman filter: linear-Gaussian models HMM: models with finite state space Stat. approx. methods for non-parametric distributions and large discrete DBN Diff. names: Sequential Monte Carlo (Handschin and Mayne 1969, Akashi and Kumamoto 1975) Particle filtering (Doucet et all 1997) Survival of the fittest (Kanazawa, Koller and Russell 1995) Condensation in computer vision (Isard and Blake 1996)

Outline Importance Sampling (IS) revisited Sequential IS (SIS) Particle Filtering = SIS + Resampling Dynamic Bayesian Networks A Simple example: ABC network Inference in DBN: Exact inference Pure Particle Filtering Rao-Blackwellised PF Demonstration in ABC network Discussions

Importance Sampling Revisited Goal: evaluate the following functional Importance Sampling (batch mode): Sample from Assign as weight of each sample The posterior estimation of is:

How to make it sequential? Choose Importance function to be: We get the SIS filter Benefit of SIS Observation y k don’t have be given in batch Sequential Importance Sampling

Sequential Importance Sampling (2)

Resampling Why need to resample Degeneracy of SIS The variance of the importance weights (y 0:k is r.v.) increases in each recursion step Optimal importance function Need to sample from and evaluate Resampling: eliminate small weights and concentrate on large weights

Proof of the Opti. IF

Resampling (2) Measure of degeneracy: effective sample size

Resampling Step Particle filtering = SIS + Resampling

Rao-Blackwellisation for SIS A method to reduce the variance of the final posterior estimation Useful when the state can be partitioned as in which can be analytically marginalized. Assuming can be evaluated analytically given, one can rewrite the posterior estimation as

Rao-Blackwellisation for SIS (2)

Rao-Blackwellisation for SIS (3) Use importance function: The estimation is given by: where

Example: ABC network

Inference in DBN n

Exact inference in ABC network

Particle filtering

Rao-Blackwellised PF

Rao-Blackwellised PF (2)

Rao-Blackwellised PF (3)

Rao-Blackwellised PF (4)

Discussions Structure of the network: A, C dependent on B y t can be also separated into 3 indep. parts