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SampleSearch Scheme for Consistent Samples: Importance Sampling & Rejection Problem

Explore a scheme that searches for consistent samples in Bayesian networks using importance sampling. Learn about the rejection problem and experimental results. The algorithm, sampling distribution, and approximation are discussed.

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SampleSearch Scheme for Consistent Samples: Importance Sampling & Rejection Problem

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  1. SampleSearch: A scheme that searches for Consistent Samples Vibhav Gogate and Rina Dechter University of California, Irvine USA

  2. Outline • Background • Bayesian Networks with Zero probabilities • Importance Sampling • Rejection Problem • The SampleSearch Scheme • Algorithm • Sampling Distribution and its Approximation • Experimental Results

  3. P(S) P(C|S) P(B|S) P(X|C,S) P(D|C,B) Bayesian Networks: Representation(Pearl, 1988) Smoking lung Cancer Bronchitis X-ray Dyspnoea P(S, C, B, X, D)= P(S) P(C|S) P(B|S) P(X|C,S) P(D|C,B) (A) Probability of Evidence P(smoking=no, dyspnoea=yes)=? (B) Belief Updating: P (lung cancer=yes | smoking=no, dyspnoea=yes ) = ?

  4. Complexity • Belief Updating • NP-hard when zeros are present • General case when all CPTs are positive, not known. • Relative Approximation • Randomized Polynomial time algorithm when all CPTs are positive (Dagum and Luby 1997) • Probability of Evidence • NP-hard when zeros are present • Relative Approximation • Randomized Polynomial time algorithm when all CPTs are positive and (1/P(e)) is polynomial (Karp, Dagum and Luby 1993)

  5. Importance Sampling (Rubinstein ’81)

  6. Importance Sampling for Belief Updating

  7. Generating i.i.d. samples from Q Q(A,B,C)=Q(A)*Q(B|A)*Q(C|A,B) Q(A)=(0.8,0.2) Q(B|A)=(0.4,0.6,0.2,0.8) Q(C|A,B)=Q(C)=(0.2,0.8) Root 0.8 0.2 A=1 A=0 0.8 0.2 0.4 0.6 B=0 B=1 B=0 B=1 0.8 0.8 0.2 0.2 0.8 0.2 0.8 0.2 C=0 C=1 C=0 C=1 C=0 C=1 C=0 C=1

  8. Rejection Problem • Importance Sampling requirement • f(xi)>0 => Q(xi)>0 • Conversely, Q(xi) can be >0 even if f(xi)=0. • So if the probability of sampling ∑Q(xi|f(xi)>0) is very small • A large number of assignments will have zero weight • Extreme case: Our approximation = zero.

  9. Rejection Problem Root All Blue leaves correspond to solutions i.e. f(x) >0 All Red leaves correspond to non-solutions i.e. f(x)=0 0.8 0.2 A=1 A=0 0.8 0.2 0.4 0.6 B=0 B=1 B=0 B=1 0.8 0.8 0.2 0.2 0.8 0.2 0.8 0.2 C=0 C=1 C=0 C=1 C=0 C=1 C=0 C=1

  10. E A D B F G C Constraint Networks(Dechter 2003) Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (red, green, blue) Constraints: A Solution is an assignment that satisfies all constraints

  11. Constraint networks to model “zeros” Constraints A=0, C=0 not allowed A=1, C=1 not allowed Or A≠C A B C • Why constraints? • For a partial sample if a constraint is violated f(X=x)=0 for any full extension X=x of the sample. • For every full assignment X=x • solution implies f(X=x) >0 and • non-solution f(X=x)=0 F D G

  12. Using Constraints Constraints A≠B, A≠C Root 0.8 0.2 A=1 A=0 0.8 0.2 0.4 0.6 B=0 B=1 B=0 B=1 0.8 0.8 0.2 0.2 0.8 0.2 0.8 0.2 C=0 C=1 C=0 C=1 C=0 C=1 C=0 C=1

  13. Using Constraints Root Constraints A≠B, A≠C 0.8 A=0 0.4 0.6 B=0 B=1 Constraint A≠B violated 0.8 0.2 0.2 0.8 C=0 C=1 C=0 C=1 C=0

  14. Outline • Background • Bayesian Networks • Importance Sampling • Rejection Prblem • The SampleSearch Scheme • Algorithm • Sampling Distribution and Approximation • Experimental Results

  15. Algorithm SampleSearch Root Constraints A≠B, A≠C 0.8 0.2 A=1 A=0 0.8 0.2 0.4 0.6 B=0 B=1 B=0 B=1 0.8 0.8 0.2 0.2 0.8 0.2 0.8 0.2 C=0 C=1 C=0 C=1 C=0 C=1 C=0 C=1

  16. Algorithm SampleSearch Root Constraints A≠B, A≠C 0.8 0.2 A=1 A=0 0.8 0.2 0.4 0.6 1 B=0 B=1 B=0 B=1 0.8 0.8 0.2 0.2 0.8 0.2 0.8 0.2 C=0 C=1 C=0 C=1 C=0 C=1 C=0 C=1

  17. Algorithm SampleSearch Root Constraints A≠B, A≠C 0.8 0.2 A=1 A=0 0.8 0.2 1 B=1 B=0 B=1 0.8 0.2 0.8 0.2 0.8 0.2 C=0 C=1 C=0 C=1 C=0 C=1 Resume Sampling

  18. Algorithm SampleSearch Root Constraints A≠B, A≠C 0.8 0.2 A=1 A=0 0.8 0.2 1 B=1 B=0 B=1 0.8 0.2 0.8 1 0.2 0.8 0.2 C=0 C=1 C=0 C=1 C=0 C=1 Constraint Violated Until Solution i.e. f(x)>0 found

  19. Generate more Samples Root Constraints A≠B, A≠C 0.8 0.2 A=1 A=0 0.8 0.2 0.4 0.6 B=0 B=1 B=0 B=1 0.8 0.8 0.2 0.2 0.8 0.2 0.8 0.2 C=0 C=1 C=0 C=1 C=0 C=1 C=0 C=1

  20. Generate more Samples Root Constraints A≠B, A≠C 0.8 0.2 A=1 A=0 0.8 0.2 0.4 0.6 B=0 B=1 B=0 B=1 0.8 1 0.8 0.2 0.2 0.8 0.2 0.8 0.2 C=0 C=1 C=0 C=1 C=0 C=1 C=0 C=1

  21. Constraints A≠B, A≠C Root Root Root A=0 A=0 A=0 B=0 B=1 B=1 B=1 B=0 C=0 C=1 C=0 C=1 C=1 Traces of SampleSearch Root A=0 B=1 C=1

  22. The Sampling distribution QR of SampleSearch • Did you generate samples from Q? -NO! Root What is probability of generating A=0? QR(A=0)=0.8 Why? SampleSearch is systematic 0.8 What is probability of generating B=1? QR(B=1|A=0)=1 Why? SampleSearch is systematic A=0 0 1 B=0 B=1 What is probability of generating B=0? Simple: QR(B=0|A=0)=0 All samples generated by SampleSearch are solutions 0 0 0 1 C=0 C=1 C=0 C=1 Backtrack-free distribution

  23. Computing QR • Invoke an oracle or a complete search procedure O(n) times per sample Root ?? Solution A=0 B=1 ?? Solution ?? Solution C=1

  24. Approximation AR of QR Root • IF Hole THEN AR=Q • IF No solutions on the other branch THEN AR=1 0.8 Hole Don’t know A=0 0 1 No solutions here B=0 B=1 0 0 0 1 No solutions here C=0 C=1 C=0 C=1

  25. Root Root Root 0.8 0.8 A=0 A=0 A=0 1 0.6 B=0 B=1 ? B=1 B=1 B=0 1 0.8 ? C=0 C=1 C=0 C=1 C=1 Approximation AR of QR Root • Problem: Can’t guarantee convergence 0.8 0.8 A=0 0.6 1 ? B=1 1 0.8 ? C=1

  26. Root Root Root 0.8 0.8 0.8 A=0 A=0 ? 1 0.6 A=0 B=1 ? B=1 1 B=0 1 0.8 B=1 ? C=0 C=1 C=1 1 C=1 Guarantee convergence in the limit • Store all possible traces Approximation ARN IF Hole THEN ARN=Q IF No solutions on other branch THEN ARN=1

  27. Improving Naive SampleSeach • Handle Non-binary domains • See the paper, Proof is complicated. • Better Search Strategy • Can use any state-of-the-art CSP/SAT solver e.g. minisat (Sorrenson et al 2006) • All theorems and result hold • Better Importance Function • Use output of generalized belief propagation to compute the initial importance function Q (Gogate and Dechter 2005)

  28. Experimental Results • Previous Algorithms • Likelihood weighting (LW) • Proposal=Prior • IJGP-sampling (IJGP-S) (Gogate and Dechter 2005) • Proposal=Output of generalized belief propagation • Adding SampleSearch • SampleSearch with LW (S+LW) • SampleSearch with IJGP-sampling (S+IJGP-S)

  29. Linkage BN_69

  30. Linkage BN_73

  31. Linkage BN_76

  32. Conclusions • Belief networks with zero probabilities lead to the Rejection problem in importance Sampling. • We presented a SampleSearch scheme that works with any importance sampling scheme to circumvent the Rejection Problem. • Sampling Distribution of SampleSearch is the backtrack-free distribution QR • Expensive to compute • Approximation of QR based on storing all traces that yields an asymptotically unbiased estimator • Empirically, when a substantial number of zero probabilities are present, SampleSearch based schemes dominate their pure sampling counter-parts.

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