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Probabilistic Networks

Probabilistic Networks. Chapter 14 of Dechter’s CP textbook Speaker: Daniel Geschwender April 1, 2013. Motivation. Hard & soft constraints are known with certainty How to model uncertainty? Probabilistic networks (also belief networks & Bayesian networks) handle uncertainty

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Probabilistic Networks

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  1. Probabilistic Networks Chapter 14 of Dechter’s CP textbook Speaker: Daniel Geschwender April 1, 2013 DanielG--Probabilistic Networks

  2. Motivation • Hard & soft constraints are known with certainty • How to model uncertainty? • Probabilistic networks (also belief networks & Bayesian networks) handle uncertainty • Not a ‘pure’ CSP but techniques (bucket elimination)can be adapted to work DanielG--Probabilistic Networks

  3. Overview • Background on probability • Probabilistic networks defined Section 14 • Belief assessment with bucket elimination Section 14.1 • Most probable explanation with Section 14.2 bucket elimination • Maximum a posteriori hypothesis [Dechter 96] • Complexity Section 14.3 • Hybrids of elimination and conditioning Section 14.4 • Summary DanielG--Probabilistic Networks

  4. Probability: Background • Single variable probability: P(b) probability of b • Joint probability: P(a,b) probability of a and b • Conditional probability: P(a|b)probability of a given b DanielG--Probabilistic Networks

  5. Chaining Conditional Probabilities • A joint probability of any size may be broken into conditional probabilities DanielG--Probabilistic Networks

  6. Section 14 Graphical Representation • Represented by a directed acyclic graph • Edges are causal influence of one variable to another • Direct influence: single edge • Indirect influence: path length ≥ 2 DanielG--Probabilistic Networks

  7. Section 14 Example A: D: F: G: B: C: Conditional Probability Table (CPT) DanielG--Probabilistic Networks

  8. Section 14 Belief Network Defined • Set of random variables: • Variables’ domains: • Belief network: • Directed acyclic graph: • Conditional prob. tables: • Evidence set: ,subset of instantiated variables DanielG--Probabilistic Networks

  9. Section 14 Belief Network Defined • A belief network gives a probability distribution over all variables in X • An assignment is abbreviated • is the restriction of to a subset of variables, S DanielG--Probabilistic Networks

  10. Section 14 Example A: D: F: G: B: C: Conditional Probability Table (CPT) DanielG--Probabilistic Networks

  11. Section 14 Example P(A=sp,B=1,C=0,D=0,F=0,G=0) = P(A=sp) ∙ P(B=1|A=sp) ∙ P(C=0|A=sp) ∙ P(D=0|A=sp,B=1) ∙ P(F=0|B=1,C=0) ∙ P(G=0|F=0) =0.25 ∙ 0.1∙ 0.7∙ 1.0∙ 0.4 ∙ 1.0 = 0.007 DanielG--Probabilistic Networks

  12. Section 14 Probabilistic Network: Queries • Belief assessment given a set of evidence, determine how probabilities of all other variables are affected • Most probable explanation (MPE) given a set of evidence, find the most probable assignment to all other variables • Maximum a posteriori hypothesis (MAP) assign a subset of unobserved hypothesis variables to maximize their conditional probability DanielG--Probabilistic Networks

  13. Section 14.1 Belief Assessment: Bucket Elimination • Belief Assessment Given a set of evidence, determine how probabilities of all other variables are affected • Evidence: Some possibilities are eliminated • Probabilities of unknowns can be updated • Known as belief updating • Solved by a modification of Bucket Elimination DanielG--Probabilistic Networks

  14. Section 14.1 Derivation • Similar to ELIM-OPT • Summation replaced with product • Maximization replaced by summation • x=a is the proposition we are considering • E=e is our evidence • Compute DanielG--Probabilistic Networks

  15. Section 14.1 ELIM-BEL Algorithm Takes as input a belief network along with an ordering on the variables. All known variable values are also provided as “evidence” DanielG--Probabilistic Networks

  16. Section 14.1 ELIM-BEL Algorithm Will output a matrix with probabilities for all values of x1 (the first variable in the given ordering) given the evidence. DanielG--Probabilistic Networks

  17. Section 14.1 ELIM-BEL Algorithm Sets up the buckets, one for each variable. As with other bucket elimination algorithms, the matrices start in the last bucket and move up until they are “caught” by the first bucket which is a variable in its scope. DanielG--Probabilistic Networks

  18. Section 14.1 ELIM-BEL Algorithm Go through all the buckets, last to first. DanielG--Probabilistic Networks

  19. Section 14.1 ELIM-BEL Algorithm If a bucket contains a piece of the input evidence, ignore all probabilities not associated with that variable assignment DanielG--Probabilistic Networks

  20. Section 14.1 ELIM-BEL Algorithm The scope of the generated matrix is the union of the scopes of the contained matrices and without the bucket variable, as it is projected out Consider all tuples of variables in the scopes and multiply their probabilities. When projecting out the bucket variable, sum the probabilities. DanielG--Probabilistic Networks

  21. Section 14.1 ELIM-BEL Algorithm To arrive at the output desired, a normalizing constant must be applied to make all probabilities of all values of x1 sum to 1. DanielG--Probabilistic Networks

  22. Section 14.1 Example A: D: F: G: B: C: Conditional Probability Table (CPT) DanielG--Probabilistic Networks

  23. Section 14.1 Example P(a) λC(a) A C P(c|a) λB(a,c) B λF(b,c) P(b|a) λD(b,a) F P(f|b,c) λG(f) D d=1 P(d|b,a) G g=1 P(g|f) DanielG--Probabilistic Networks

  24. Section 14.1 Example G g=1 λG(f) P(g|f) g=1 P(g|f) λG(f) DanielG--Probabilistic Networks

  25. Section 14.1 Example λD(b,a) D d=1 P(d|b,a) P(d|b,a) d=1 λD(b,a) DanielG--Probabilistic Networks

  26. Section 14.1 Example λF(b,c) F P(f|b,c) λG(f) λF(b,c) λG(f) P(f|b,c) DanielG--Probabilistic Networks

  27. Section 14.1 Example B λB(a,c) λF(b,c) P(b|a) λD(b,a) λF(b,c) λB(a,c) λD(b,a) P(b|a) DanielG--Probabilistic Networks

  28. Section 14.1 Example λC(a) C P(c|a) λB(a,c) P(c|a) λB(a,c) λC(a) DanielG--Probabilistic Networks

  29. Section 14.1 Example λA(a) P(a) λC(a) A λC(a) λA(a) P(a) Σ=0.00595 DanielG--Probabilistic Networks

  30. Section 14.1 Derivation • Evidence that g=1 • Need to compute: • Generate a function over G, DanielG--Probabilistic Networks

  31. Section 14.1 Derivation • Place as far left as possible: • Generate . Place as far left as possible. • Generate . DanielG--Probabilistic Networks

  32. Section 14.1 Derivation • Generate and place . • Generate and place . • Thus our final answer is DanielG--Probabilistic Networks

  33. Section 14.2 ELIM-MPE Algorithm As before, takes as input a belief network along with an ordering on the variables. All known variable values are also provided as “evidence”. DanielG--Probabilistic Networks

  34. Section 14.2 ELIM-MPE Algorithm The output will be the most probable configuration of the variables considering the given evidence. We will also have the probability of that configuration. DanielG--Probabilistic Networks

  35. Section 14.2 ELIM-MPE Algorithm Buckets are initialized as before. DanielG--Probabilistic Networks

  36. Section 14.2 ELIM-MPE Algorithm Iterate buckets from last to first. (Note that the functions are referred to by h rather than λ) DanielG--Probabilistic Networks

  37. Section 14.2 ELIM-MPE Algorithm If a bucket contains evidence, ignore all assignments that go against that evidence. DanielG--Probabilistic Networks

  38. Section 14.2 ELIM-MPE Algorithm The scope of the generated function is the union of the scopes of the contained functions but without the bucket variable. The function is generated by multiplying corresponding entries in the contained matrices and then projecting out the bucket variable by taking the maximum probability. DanielG--Probabilistic Networks

  39. Section 14.2 ELIM-MPE Algorithm The probability of the MPE is returned when the final bucket is processed. DanielG--Probabilistic Networks

  40. Section 14.2 ELIM-MPE Algorithm Return to all the buckets in the order d and assign the value that maximizes the probability returned by the generated functions. DanielG--Probabilistic Networks

  41. Section 14.2 Example A: D: F: G: B: C: Conditional Probability Table (CPT) DanielG--Probabilistic Networks

  42. Section 14.2 Example P(a) hC(a) A C P(c|a) hB(a,c) B hF(b,c) P(b|a) hD(b,a) F P(f|b,c) hG(f) f=1 D P(d|b,a) G P(g|f) DanielG--Probabilistic Networks

  43. Section 14.2 Example G hG(f) P(g|f) P(g|f) hG(f) DanielG--Probabilistic Networks

  44. Section 14.2 Example hD(b,a) D P(d|b,a) P(d|b,a) hD(b,a) DanielG--Probabilistic Networks

  45. Section 14.2 Example hF(b,c) F f=1 hG(f) P(f|b,c) f=1 hF(b,c) P(f|b,c) hG(f) DanielG--Probabilistic Networks

  46. Section 14.2 Example B hB(a,c) hF(b,c) P(b|a) hD(b,a) hF(b,c) hB(a,c) hD(b,a) P(b|a) DanielG--Probabilistic Networks

  47. Section 14.2 Example hC(a) C P(c|a) hB(a,c) P(c|a) hB(a,c) hC(a) DanielG--Probabilistic Networks

  48. Section 14.2 Example hA(a) P(a) hC(a) A hC(a) hA(a) P(a) max=0.02126 DanielG--Probabilistic Networks

  49. Section 14.2 Example hC(a) hB(a,c) hA(a) hF(b,c) hD(b,a) hG(f) MPE probability: 0.02126 DanielG--Probabilistic Networks

  50. Section 14.2 Example hC(a) hB(a,c) hA(a) hF(b,c) hD(b,a) hG(f) MPE probability: 0.02126 A=sp DanielG--Probabilistic Networks

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