Inference in Bayesian Networks

1. Inference in Bayesian Networks A. Darwiche

2. Pr: Evidence: Pr(e) Posterior marginals: Pr(x|e) for every X MPE: Most probable instantiation: Instantiation y such that Pr(y|e) is maximal (Y = E) MAP: Maximum a posteriori hypothesis: Intantiation y such that Pr(y|e) is maximal (Y is subset of E) Query Types A. Darwiche

3. Pr: Posterior Marginals Battery Age Alternator Fan Belt Charge Delivered Battery Fuel Pump Fuel Line Starter Distributor Gas Battery Power Spark Plugs Gas Gauge Engine Start Lights Engine Turn Over Radio A. Darwiche

4. Battery Age Alternator Fan Belt Charge Delivered Battery Fuel Pump Fuel Line Starter Distributor Gas Battery Power Spark Plugs Gas Gauge Engine Start Lights Engine Turn Over Radio ok on yes no ok off yes no Diagnosis Scenario .001 .090 A. Darwiche

5. MPE: Most Probable Explanation Battery Age Alternator Fan Belt Charge Delivered Battery Fuel Pump Fuel Line Starter Distributor Gas Battery Power Spark Plugs Gas Gauge Engine Start Lights Engine Turn Over Radio A. Darwiche

6. MPE: Most Probable Explanation Battery Age Alternator Fan Belt Charge Delivered Battery Fuel Pump Fuel Line Starter Distributor Gas Battery Power Spark Plugs Gas Gauge Engine Start Lights Engine Turn Over Radio A. Darwiche

7. MAP: Maximum a Posteriori Hypothesis Battery Age Alternator Fan Belt Charge Delivered Battery Fuel Pump Fuel Line Starter Distributor Gas Battery Power Spark Plugs Gas Gauge Engine Start Lights Engine Turn Over Radio A. Darwiche

8. MAP: Maximum a Posteriori Hypothesis Battery Age Alternator Fan Belt Charge Delivered Battery Fuel Pump Fuel Line Starter Distributor Gas Battery Power Spark Plugs Gas Gauge Engine Start Lights Engine Turn Over Radio A. Darwiche

9. MAP: Maximum a Posteriori Hypothesis Battery Age Alternator Fan Belt Charge Delivered Battery Fuel Pump Fuel Line Starter Distributor Gas Battery Power Spark Plugs Gas Gauge Engine Start Lights Engine Turn Over Radio A. Darwiche

10. Probability of Evidence A. Darwiche

11. + * * A B ØB A B + + A ØA Factoring true false .9 true .3 true .1 true false true .8 false .7 * * * * * * false false .2 .3 λa λb .7 .1 .9 .8 .2 λ~a λ~b false A. Darwiche

12. Notation • A binary variable X: • is variable with two values (true, false) • x is short notation for X=true • ~x is short notation for X=false • If X is a variable with parents Y and Z, then: represents the probability Pr(X=x | Y=y, Z=y) • If X is a binary variable with parents Y and Z (also binary), then: represents the probability Pr(X=true | Y=false, Z=true) A. Darwiche

13. An instantiation is a set of variables with their values: X=true,Y=false, Z=true is an instantiation A=a, B=b, C=c is an instantiation x, ~y, z is short notation for the instantiationX=true, Y=false, Z=true a,b,c is short notation for the instantiation A=a, B=b, C=c Two instantiations are inconsistent iff they assign different values to the same variable: x,~y,z and x,y,z are inconsistent x,~y,z and a,b,c are consistent Notation A. Darwiche

14. A B Pr true true .03 true false .27 false true .56 false false false .14 false Joint Probability Distribution Pr(a) = .03 + .27 = .3 A. Darwiche

15. A B Pr true true .03 true false .27 false true .56 false false false .14 false Joint Probability Distribution Pr(~b) = .27 + .14 = .41 A. Darwiche

16. λa λb …are called evidence indicators Evidence Indicators A B Pr λaλb .03 .03 true true .27 λaλ~b .27 true false λ~aλb .56 .56 false true false λ~aλ~b .14 .14 false false false F = .03λaλb + .27λaλ~b + .56λ~aλb + .14λ~a λ~b F is called the polynomial of the given probability distribution A. Darwiche

17. To compute the probability of instantiation e: Evaluate polynomial F while replacing each indicator -by 1 if the instantiation is consistent with the indicator;-by 0 if the instantiation is inconsistent with the indicator Examples: Indicator λa is consistent with instantiation a,~b,c Indicator λb is inconsistent with instantiation a,~b,c Indicator λd is consistent with instantiation a,~b,c Indicator λ~d is consistent with instantiation a,~b,c Computing Probabilities A. Darwiche

18. Computing Probabilities F = .03λaλb + .27λaλ~b + .56λ~aλb + .14λ~a λ~b To compute the probability of instantiation a, ~b: F(a,~b) = .03*1*0+ .27*1*1+ .56*0*0+ .14*0*1 = .27 To compute the probability of instantiation ~a: F(~a) = .03*0*1+ .27*0*1+ .56*1*1+ .14*1*1 = .70 A. Darwiche

19. A B Pr .03=.3*.1 true true A B .27=.3*.9 true false ØB A B A ØA true false .9 56=.7*.8 false true true .3 true .1 true false true .8 false .7 false false .2 false .14=.7*.2 false false false A. Darwiche

20. A B ØB A B A ØA θ b|a true true θa θ~b|a true true false θ~a θ b|~a false true false θ ~b|~a false false A B Pr θa θ b|a true true θa θ~b|a true false θ~a θ b|~a false true false θ~a θ ~b|~a false false false A. Darwiche

21. A B ØB A B A ØA θ b|a true true θa θ~b|a true true false θ~a θ b|~a false true false θ ~b|~a false false A B Pr λaλbθa θ b|a true true λaλ~bθa θ~b|a true false λ~aλbθ~a θ b|~a false true false λ~aλ~bθ~a θ ~b|~a false false false A. Darwiche

22. A B ØB A B A ØA θ b|a true true θa θ~b|a true true false θ~a θ b|~a false true false θ ~b|~a false false A B Pr λaλbθa θ b|a true true λaλ~bθa θ~b|a true false λ~aλbθ~a θ b|~a false true false λ~aλ~bθ~a θ ~b|~a false false λa λb θa θb|a+ λa λ~b θa θ~b|a+ λ~aλb θ~a θb|~a+ λ~a λ~b θ~a θ~b|~a A. Darwiche

23. B A ØB A B A ØA true true θ b|a true θa true false θ~b|a false θ~a false true θ b|~a false false θ ~b|~a false The Polynomial of a Bayesian Network λa λb θa θb|a+ λa λ~b θa θ~b|a+ λ~aλb θ~a θb|~a+ λ~a λ~b θ~a θ~b|~a A. Darwiche

24. C B D A The Polynomial of a Bayesian Network F = λa λb λc λd θa θb|a θc|a θd|bc + λa λb λc λ~d θa θb|a θc|a θ~d|bc + …. A. Darwiche

25. + Factoring * * + + * * * * * * θa λa λb θab θa~b θ~ab λ~b θ~a~b λ~a θ~a Arithmetic Circuit λa λb θa θb|a+ λa λ~b θa θ~b|a+ λ~aλb θ~a θb|~a+ λ~a λ~b θ~a θ~b|~a A. Darwiche

26. Arithmetic Circuit Pr(a) .3 + * * .3 0 + + 1 1 * * * * * .3 .1 .9 .8 .2 0 * θa λa λb 1 1 1 0 .3 .1 .9 .8 .2 .7 θab θa~b θ~ab λ~b θ~a~b λ~a θ~a A. Darwiche

27. + * * A B ØB A B + + A ØA Factoring true false .9 true .3 true .1 true false true .8 false .7 * * * * * * false false .2 θa λa λb θ~a θb|a θ~b|a θb|~a θ~b|~a λ~a λ~b false A. Darwiche

28. T S1 S2 S3 … Sn Factoring the Polynomial of a Bayesian Network A. Darwiche

29. Sophisticated Platform (desktop) Eval Eval Eval compiler A. Circuit A. Circuit A. Circuit Primitive Platforms (embedded) Embedding Probabilistic Reasoning Systems A. Darwiche

30. TreeWidth(Measures connectivity of Networks) Higher treewidth A. Darwiche

31. TreeWidth(Measures connectivity of Networks) Multiply-connected networks Singly-connected network (polytree) A. Darwiche

32. The treewidth of a polytree is m, where m is the maximum number of parents that any node If each node has at most one parent, the polytree is called a tree The treewidth of a tree is 1 Treewidth A. Darwiche

33. Given a Bayesian network N with: Number of nodes: n Treewith: w We can generate an arithmetic circuit for N: In O(n 2w) time In O(n 2w) space It is easy to do inference on polytrees Treewidth A. Darwiche