Bayesian Statistics and Belief Networks

1. Bayesian Statistics and Belief Networks

2. Overview • Book: Ch 8.3 • Refresher on Bayesian statistics • Bayesian classifiers • Belief Networks / Bayesian Networks

3. Why Should We Care? • Theoretical framework for machine learning, classification, knowledge representation, analysis • Bayesian methods are capable of handling noisy, incomplete data sets • Bayesian methods are commonly in use today

4. Bayesian Approach To Probability and Statistics • Classical Probability : Physical property of the world (e.g., 50% flip of a fair coin). True probability. • Bayesian Probability : A person’s degree of belief in event X. Personal probability. • Unlike classical probability, Bayesian probabilities benefit from but do not require repeated trials - only focus on next event; e.g. probability Seawolves win next game?

5. Bayes Rule Product Rule: Equating Sides: i.e. All classification methods can be seen as estimates of Bayes’ Rule, with different techniques to estimate P(evidence|Class).

6. Simple Bayes Rule Example Probability your computer has a virus, V, = 1/1000. If virused, the probability of a crash that day, C, = 4/5. Probability your computer crashes in one day, C, = 1/10. P(C|V)=0.8 P(V)=1/1000 P(C)=1/10 Even though a crash is a strong indicator of a virus, we expect only 8/1000 crashes to be caused by viruses. Why not compute P(V|C) from direct evidence? Causal vs. Diagnostic knowledge; (consider if P(C) suddenly drops).

7. Bayesian Classifiers If we’re selecting the single most likely class, we only need to find the class that maximizes P(e|Class)P(Class). Hard part is estimating P(e|Class). Evidence e typically consists of a set of observations: Usual simplifying assumption is conditional independence:

8. Bayesian Classifier Example Probability C=Virus C=Bad Disk P(C) 0.4 0.6 P(crashes|C) 0.1 0.2 P(diskfull|C) 0.6 0.1 Given a case where the disk is full and computer crashes, the classifier chooses Virus as most likely since (0.4)(0.1)(0.6) > (0.6)(0.2)(0.1).

9. Beyond Conditional Independence • Include second-order dependencies; i.e. pairwise combination of variables via joint probabilities: Linear Classifier: C1 C2 Correction factor - Difficult to compute - joint probabilities to consider

10. Belief Networks • DAG that represents the dependencies between variables and specifies the joint probability distribution • Random variables make up the nodes • Directed links represent causal direct influences • Each node has a conditional probability table quantifying the effects from the parents • No directed cycles

11. Burglary Alarm Example P(B) P(E) Burglary Earthquake 0.001 0.002 B E P(A) T T 0.95 Alarm T F 0.94 F T 0.29 F F 0.001 A P(J) A P(M) John Calls Mary Calls T 0.70 T 0.90 F 0.01 F 0.05

12. Sample Bayesian Network

13. Using The Belief Network P(B) P(E) Burglary Earthquake 0.002 0.001 B E P(A) T T 0.95 Alarm T F 0.94 F T 0.29 F F 0.001 A P(M) JohnCalls Mary Calls T 0.70 A P(J) F 0.01 T 0.90 F 0.05 Probability of alarm, no burglary or earthquake, both John and Mary call:

14. Belief Computations • Two types; both are NP-Hard • Belief Revision • Model explanatory/diagnostic tasks • Given evidence, what is the most likely hypothesis to explain the evidence? • Also called abductive reasoning • Belief Updating • Queries • Given evidence, what is the probability of some other random variable occurring?

15. Belief Revision • Given some evidence variables, find the state of all other variables that maximize the probability. • E.g.: We know John Calls, but not Mary. What is the most likely state? Only consider assignments where J=T and M=F, and maximize. Best:

16. Belief Updating • Causal Inferences • Diagnostic Inferences • Intercausal Inferences • Mixed Inferences E Q Q E Q E E Q E

17. Causal Inferences P(B) P(E) Burglary Earthquake Inference from cause to effect. E.g. Given a burglary, what is P(J|B)? 0.002 0.001 B E P(A) T T 0.95 Alarm T F 0.94 F T 0.29 F F 0.001 A P(M) JohnCalls Mary Calls T 0.70 A P(J) F 0.01 T 0.90 F 0.05 P(M|B)=0.67 via similar calculations

18. Diagnostic Inferences From effect to cause. E.g. Given that John calls, what is the P(burglary)? What is P(J)? Need P(A) first: Many false positives.

19. Intercausal Inferences Explaining Away Inferences. Given an alarm, P(B|A)=0.37. But if we add the evidence that earthquake is true, then P(B|A^E)=0.003. Even though B and E are independent, the presence of one may make the other more/less likely.

20. Mixed Inferences Simultaneous intercausal and diagnostic inference. E.g., if John calls and Earthquake is false: Computing these values exactly is somewhat complicated.

21. Exact Computation - Polytree Algorithm • Judea Pearl, 1982 • Only works on singly-connected networks - at most one undirected path between any two nodes. • Backward-chaining Message-passing algorithm for computing posterior probabilities for query node X • Compute causal support for X, evidence variables “above” X • Compute evidential support for X, evidence variables “below” X

22. Polytree Computation ... U(1) U(m) X Z(1,j) Z(n,j) ... Y(1) Y(n) Algorithm recursive, message passing chain

23. Other Query Methods • Exact Algorithms • Clustering • Cluster nodes to make single cluster, message-pass along that cluster • Symbolic Probabilistic Inference • Uses d-separation to find expressions to combine • Approximate Algorithms • Select sampling distribution, conduct trials sampling from root to evidence nodes, accumulating weight for each node. Still tractable for dense networks. • Forward Simulation • Stochastic Simulation

24. Summary • Bayesian methods provide sound theory and framework for implementation of classifiers • Bayesian networks a natural way to represent conditional independence information. Qualitative info in links, quantitative in tables. • NP-complete or NP-hard to compute exact values; typical to make simplifying assumptions or approximate methods. • Many Bayesian tools and systems exist

25. References • Russel, S. and Norvig, P. (1995). Artificial Intelligence, A Modern Approach. Prentice Hall. • Weiss, S. and Kulikowski, C. (1991). Computer Systems That Learn. Morgan Kaufman. • Heckerman, D. (1996). A Tutorial on Learning with Bayesian Networks. Microsoft Technical Report MSR-TR-95-06. • Internet Resources on Bayesian Networks and Machine Learning: http://www.cs.orst.edu/~wangxi/resource.html