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

Xt Xt+1

Yt Yt+1

X0 X1 X2

Y0 Y1 Y2

Unrolled DBN for t=0 to t=10

X10

Y10

Dynamic Belief Networks (DBNs)

Two-stage influence diagram Interaction graph

Notation

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}

X0 X1 X2

Y0 Y1 Y2

DBN

t=0 t=1 t=2

Xt Xt+1

Yt Yt+1

t=1 t=2

2-time slice

Inference is hard, need approximationMini-bucket? Sampling?

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

Particlet={at,bt,ct}

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

• 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

Example

Sample

Only Bt