1. Particle Filtering
ICS 275b
2002

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

3. Notation Xt – value of X at time tY0 Y1 Y2 Yt
Yt+1
DBN
2-time slice
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}

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 zt and observed variables yt
Objective:
or filtering density
To solve this problem,one need approximation schemes because of intractable integrals

11. Rao-Blackwellised PF Assume conditional posterior distribution
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