Importance Sampling

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Importance Sampling

• ICS 276
• Fall 2007
• Rina Dechter

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Outline

• Gibbs Sampling
• Advances in Gibbs sampling
• Blocking
• Cutset sampling (Rao-Blackwellisation)
• Importance Sampling
• Advances in Importance Sampling
• Particle Filtering

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Importance Sampling Theory
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Importance Sampling Theory

• Given a distribution called the proposal
  distribution Q (such that P(Zz,e)gt0gt Q(Zz)gt0)

w(Zz) is called as importance weight
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Importance Sampling Theory
Underlying principle, Approximate Average over a
set of numbers by an average over a set of
sampled numbers
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Importance Sampling (Informally)

• Express the problem as computing the average over
  a set of real numbers
• Sample a subset of real numbers
• Approximate the true average by sample average.
• True Average
• Average of (0.11, 0.24, 0.55, 0.77,
  0.88,0.99)0.59
• Sample Average over 2 samples
• Average of (0.24, 0.77) 0.505

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How to generate samples from Q

• Express Q in product form
• Q(Z)Q(Z1)Q(Z2Z1).Q(ZnZ1,..Zn-1)
• Sample along the order Z1,..Zn
• Example
• Q(Z1)(0.2,0.8)
• Q(Z2Z1)(0.2,0.8,0.1,0.9)
• Q(Z3Z1,Z2)Q(Z3Z1)(0.5,0.5,0.3,0.7)

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How to sample from Q
Q(Z1)(0.2,0.8) Q(Z2Z1)(0.2,0.8,0.1,0.9) Q(Z3Z1
,Z2)Q(Z3Z1)(0.5,0.5,0.3,0.7)
Domains of each variable is 0,1

• Generate a random number between 0 and 1

0
1
0.2
Which value to select for Z1?
1
0
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How to sample from Q?

• Each Sample Zz
• Sample Z1z1 from Q(Z1)
• Sample Z2z2 from Q(Z2Z1z1)
• Sample Z3z3 from Q(Z3Z1z1)
• Generate N such samples

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Likelihood weighting

• Q Prior DistributionCPTs of the Bayesian network

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Likelihood weighting example
P(S)
Smoking
P(BS)
P(CS)
lung Cancer
Bronchitis
P(XC,S)
P(DC,B)
X-ray
Dyspnoea
P(S, C, B, X, D) P(S) P(CS) P(BS) P(XC,S)
P(DC,B)
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Likelihood weighting example
P(S)
Smoking
QPrior Q(S,C,D)Q(S)Q(CS)Q(DC,B0) P(S)P(CS
)P(DC,B0)
P(BS)
P(CS)
lung Cancer
Bronchitis
Sample Ss from P(S) Sample Cc from
P(CSs) Sample Dd from P(DCc,B0)
P(XC,S)
P(DC,B)
X-ray
Dyspnoea
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The Algorithm
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How to solve belief updating?
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Difference between estimating P(Ee) and
P(XixiEe)
Unbiased
Asymptotically Unbiased
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Proposal Distribution Which is better?
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Outline

• Gibbs Sampling
• Advances in Gibbs sampling
• Blocking
• Cutset sampling (Rao-Blackwellisation)
• Importance Sampling
• Advances in Importance Sampling
• Particle Filtering

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Research Issues in Importance Sampling

• Better Proposal Distribution
• Likelihood weighting
• Fung and Chang, 1990 Shachter and Peot, 1990
• AIS-BN
• Cheng and Druzdzel, 2000
• Iterative Belief Propagation
• Changhe and Druzdzel, 2003
• Iterative Join Graph Propagation and variable
  ordering
• Gogate and Dechter, 2005

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Research Issues in Importance Sampling (Cheng and
Druzdzel 2000)

• Adaptive Importance Sampling

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Adaptive Importance Sampling

• General case
• Given k proposal distributions
• Take N samples out of each distribution
• Approximate P(e)

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Estimating Q'(z)
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Cutset importance sampling
(Gogate and Dechter, 2005) and (Bidyuk and
Dechter 2006)

• Divide the Set of variables into two parts
• Cutset (C) and Remaining Variables (R)

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Outline

• Gibbs Sampling
• Advances in Gibbs sampling
• Blocking
• Cutset sampling (Rao-Blackwellisation)
• Importance Sampling
• Advances in Importance Sampling
• Particle Filtering

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Dynamic Belief Networks (DBNs)
Transition arcs
Xt
Xt1
Yt
Yt1
Bayesian Network at time t
Bayesian Network at time t1
X10
X0
X1
X2
Y10
Y0
Y1
Y2
Unrolled DBN for t0 to t10
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Query

• Compute P(X 0t Y 0t ) or P(X t Y 0t )
• Example P(X010Y010) or P(X10Y010)
• 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
On white board