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Particle Filtering ICS 275b 2002
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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
Y0 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}
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Query Compute P(X 0:t |Y 0:t ) or P(X t |Y 0:t )
Hard!!! over a long time period Approximate! Sample!
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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 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
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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
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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:
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Particle Filtering and Rao-Blackwellisation
Monte Carlo integration
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