Dynamic Bayesian Networks and Particle Filtering
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Dynamic Bayesian Networks and Particle Filtering
COMPSCI 276(chapter 15, Russel and Norvig)
2007
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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
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Dynamic Belief Networks (DBNs)
Two-stage influence diagram Interaction graph
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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
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Inference is hard, need approximationMini-bucket? Sampling?
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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.
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Particle Filtering
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Example
Particlet={at,bt,ct}
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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
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• 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
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Example
Sample
Only Bt
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