1. A. Darwiche Bayesian Networks

2. A. Darwiche Bayesian Network Battery Age Alternator Fan Belt Battery Charge Delivered Battery Power Starter Radio LightsEngine Turn Over Gas Gauge Gas Fuel Pump Fuel Line Distributor Spark Plugs Engine Start

3. A. Darwiche Bayesian Network Battery Age Alternator Fan Belt Battery Charge Delivered Battery Power Starter Radio LightsEngine Turn Over Gas Gauge Gas Fuel Pump Fuel Line Distributor Spark Plugs Engine Start Pr(Lights=ON | Battery-Power=OK) =.99 ON OFF OK WEAK DEAD Lights Battery Power.99.01.20.80 01.99 θ 1 + θ 2 = 1

4. A. Darwiche

8. A Bayesian Network Compact representation of a probability distribution: –Complete model –Consistent model Embeds many independence assumptions: –Faithful model

9. A. Darwiche

13. A Bayesian Network Compact representation of a probability distribution: –Complete model –Consistent model Embeds many independence assumptions: –Faithful model

14. A. Darwiche Bayesian Network Earthquake (E) Burglary (B) Alarm (A) Pr(E=true)Pr(E=false).1.9 Pr(B=true)Pr(B=false).2.8 Pr(A=true)Pr(A=false) E=true, B=true.95.05 E=false, B=true.9.1 E=true, B=false.7.3 E=false, B=false.01.99

15. A. Darwiche Joint Probability Distribution EBAPr(.) True.019 True False.001 TrueFalseTrue.056 TrueFalse.024 FalseTrue.162 FalseTrueFalse.018 False True.0072 False.7128

16. A. Darwiche Independence Assumptions of a Bayesian Network

17. A. Darwiche Chol Test1Test2 Causal Structure I(Test1,Test2 | Chol)

18. A. Darwiche Chol Test1Test2 Causal Structure Nurse I(Test1,Test2 | Chol, Nurse) I(Test1,Test2 | Chol)

19. A. Darwiche H O1On Naïve Bayes O2 H: Disease O1, …, On: Findings (symptoms, lab tests, …) …

20. A. Darwiche Genetic Tracking G1G2 G3G4G5 G6 G7G8P4 Each node is independent of its non-descendants given its parents

21. A. Darwiche Genetic Tracking G1G2 G3G4G5 G6 G7G8P4 Each node is independent of its non-descendants given its parents

22. A. Darwiche Genetic Tracking G1G2 G3G4G5 G6 G7G8P4 Each node is independent of its non-descendants given its parents

23. A. Darwiche Dynamic Systems S1 O1 S2 O2 S3 O3 S4 O4 S5 O5 Each node is independent of its non-descendants given its parents

24. A. Darwiche Dynamic Systems S1 O1 S2 O2 S3 O3 S4 O4 S5 O5 Each node is independent of its non-descendants given its parents

25. A. Darwiche The chain rule for Bayesian Networks

26. A. Darwiche Earthquake (E) Burglary (B) Alarm (A) Call (C) Radio (R) Pr(c|a) Pr(craeb)=Pr(c|raeb)Pr(r|aeb)Pr(a|eb)Pr(e|b)Pr(b) Pr(r|e)Pr(a|eb)Pr(e)Pr(b) Pr(e)Pr(b) Pr(a|eb) Pr(r|e) Pr(c|a)

27. A. Darwiche Example: Build Joint Probability Table Earthquake (E) Burglary (B) Alarm (A) Pr(E=true)Pr(E=false).1.9 Pr(B=true)Pr(B=false).2.8 Pr(A=true)Pr(A=false) E=true, B=true.95.05 E=false, B=true.9.1 E=true, B=false.7.3 E=false, B=false.01.99

28. A. Darwiche Temperature/Sensors Temperature: high (20%), low (10%), nominal (70%) 3 Sensors (true, false): true (90%) given high temperature true (1%) given low temperature true (5%) given nominal temperature

29. A. Darwiche

32. Queries Pr(Sensor1=true)? Pr(Temperature=high | Sensor1=true)? Pr(Temperature=high | Sensor1=true, Sensor2=true, Sensor3=true)?

33. A. Darwiche d-separation

34. Earthquake (E) Burglary (B) Alarm (A) Call (C) Radio (R) … (F) Is A Independent of R given E?

35. A. Darwiche Earthquake (E) Burglary (B) Alarm (A) Call (C) Radio (R) Chain Link E & C not d-separated …Active!

36. A. Darwiche Earthquake (E) Burglary (B) Alarm (A) Call (C) Radio (R) Chain Link E & C are d-separated by A …Blocked!

37. A. Darwiche Earthquake (E) Burglary (B) Alarm (A) Call (C) Radio (R) Divergent Link R & A not d-seperated …Active!

38. A. Darwiche Earthquake (E) Burglary (B) Alarm (A) Call (C) Radio (R) Divergent Link R & A d-separated by E …Blocked!

39. A. Darwiche Earthquake (E) Burglary (B) Alarm (A) Call (C) Radio (R) Convergent Link E & B d-seperated …Blocked!

40. A. Darwiche Earthquake (E) Burglary (B) Alarm (A) Call (C) Radio (R) Convergent Link E & B not d-separated by A …Active!

41. A. Darwiche Earthquake (E) Burglary (B) Alarm (A) Call (C) Radio (R) Convergent Link E & B not d-separated by C …Active!

42. A. Darwiche Earthquake (E) Burglary (B) Alarm (A) Call (C) Radio (R) Are B & R d-separated by E & C ? Active Blocked

43. A. Darwiche Earthquake (E) Burglary (B) Alarm (A) Call (C) Radio (R) Active Are C & R d-separated ?

44. A. Darwiche blocked active

45. A. Darwiche d-separation Nodes X are d-separated from nodes Y by nodes Z iff every path from X to Y is blocked by Z. A path is blocked by Z if some link on the path is blocked: –For some →X→ or ←X→, X in Z –For some →X←, neither X nor one of its descendents in Z

46. A. Darwiche d-separation in Asia Network Visit to Asia / Smoker: –No evidence: No –Given TB-or-Cancer: Yes –Given +ve X-Ray: Yes Visit to Asia / +ve X-ray: –No evidence: Yes –Given TB: No –Given TB-or-Cancer: No Bronchitis / Lung Cancer: –No evidence: Yes –Given Smoker: No –Given Smoker and Dysnpnoea: Yes