Particle Filtering

1. Particle Filtering ICS 275b 2002

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

3. Notation t=0 t=1 t=2 t=1 t=2 X0 X1 X2 Xt Xt+1 Xt – value of X at time t X 0:t ={X0,X1,…,Xt}– vector of values of X Yt – evidence at time t Y 0:t = {Y0,Y1,…,Yt} Y0 Y1 Y2 Yt Yt+1 DBN 2-time slice

4. Query • Compute P(X 0:t |Y 0:t ) or P(X t |Y 0:t ) • Hard!!! over a long time period • Approximate! Sample!

5. 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.

6. Particle Filtering

7. Example Particlet={at,bt,ct}

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

9. Drawback of PF • 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

10. Problem Formulation • Model : general state space model/DBN with hidden variables ztand observed variables yt • Objective: • or filtering density • To solve this problem,one need approximation schemes because of intractable integrals

11. Rao-Blackwellised PF • Divide hidden variables into two groups: rt and xt • Assume conditional posterior distribution p(x0:t | y1:t ,r0:t ,) is analytically tractable • We only need to focus on estimating p(r0:t | y1:t), which lies in a space of reduced dimension:

12. Particle Filtering and Rao-Blackwellisation • Monte Carlo integration