Dynamic Bayesian Networks and Particle Filtering COMPSCI 276 (chapter 15, Russel and Norvig) 2007

Dynamic Belief Networks (DBNs) Bayesian Network at time t Bayesian Network at time t+1 Transition arcs XtXt X t+1 YtYt Y t+1 X0X0 X1X1 X2X2 Y0Y0 Y1Y1 Y2Y2 Unrolled DBN for t=0 to t=10 X 10 Y 10

Dynamic Belief Networks (DBNs) Two-stage influence diagram Interaction graph

Notation X t – value of X at time t X 0:t ={X 0,X 1,…,X t }– vector of values of X Y t – evidence at time t Y 0:t = {Y 0,Y 1,…,Y t } X0X0 X1X1 X2X2 Y0Y0 Y1Y1 Y2Y2 DBN t=0 t=1t=2 XtXt X t+1 YtYt Y t+1 t=1t=2 2-time slice

Inference is hard, need approximation Mini-bucket? Sampling?

Same Queries Compute P(X 0:t |Y 0:t ) or P(X t |Y 0:t ) –Example P(X 0:10 |Y 0:10 ) or P(X 10 |Y 0:10 ) –Filtering –Prediction –Smoothing –MPE Hard!!! over a long time period Approximate! Sample!

Particle Filtering (PF) = “ condensation ” = “ sequential Monte Carlo ” = “ survival of the fittest ” –PF can treat any type of probability distribution, non-linearity, and non-stationarity; –PF are powerful sampling based inference/learning algorithms for DBNs.

Particle Filtering

Example Particle t ={a t,b t,c t }

PF Sampling Particle (t) ={a t,b t,c t } Compute particle (t+1): Sample b t+1, from P(b|a t,c t ) Sample a t+1, from P(a|b t+1,c t ) Sample c t+1, from P(c|b t+1,a t+1 ) Weight particle w t+1 If weight is too small, discard Otherwise, multiply

Drawback of PF –Inefficient 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 Drawback of PF

Problem Formulation Model : general state space model/DBN with hidden variables z t and observed variables y t Objective: –or filtering density –To solve this problem,one need approximation schemes because of intractable integrals

Assume conditional posterior distribution p(x 0:t | y 1:t,r 0:t,) is analytically tractable We only need to focus on estimating p(r 0:t | y 1:t ), which lies in a space of reduced dimension: Rao-Blackwellised PF Divide hidden variables into two groups: r t and x t

Example Sample Only B t

Importance Sampling and Rao- Blackwellisation Monte Carlo integration