A Brief Introduction to Bayesian Networks AIMA CIS 391 Intro to Artificial Intelligence
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1 A Brief Introduction to Bayesian Networks AIMA CIS 391 Intro to Artificial Intelligence (LDA slides from Lyle Ungar from slides by Jonathan Huang
2 Bayesian networks A simple, graphical notation for conditional independence assertions and hence for compact specification of full joint distributions Syntax: A set of nodes, one per random variable A set of directed edges (link "directly influences"), yielding a directed acyclic graph A conditional distribution for each node given its parents: P (X i Parents (X i )) In the simplest case, conditional distribution represented as a conditional probability table (CPT) giving the distribution over X i for each combination of parent values CIS Intro to AI 2
3 The Required-By-Law Burglar Example I'm at work, and my neighbor John calls, but my neighbor Mary doesn't call. They usually call if my burglar alarm goes off, and occasionally otherwise. Sometimes the alarm is set off by minor earthquakes. Is there a burglar? Variables: Burglary, Earthquake, Alarm, JohnCalls, MaryCalls Network topology reflects causal knowledge: 1. A burglar can cause the alarm to go off 2. An earthquake can cause the alarm to go off 3. The alarm can cause Mary to call 4. The alarm can cause John to call CIS Intro to AI 3
4 Belief network for example Network topology reflects "causal" knowledge: 1. A burglar can cause the alarm to go off 2. An earthquake can cause the alarm to go off 3. The alarm can cause Mary to call (with prob.70) 4. The alarm can cause John to call (with prob.90) CIS Intro to AI 4
5 Belief network for example Network topology reflects "causal" knowledge: 1. A burglar can cause the alarm to go off (with probability.94) 2. An earthquake can cause the alarm to go off (with prob.29) 3. The alarm can cause Mary to call 4. The alarm can cause John to call CIS Intro to AI 5
6 Belief network for example Unconditional probabilities reflect priors of those events: 1. The probability of a burglary (per day?) is The probability of an earthquake is.002 CIS Intro to AI 6
7 Belief network with all (conditional) probability tables Makes it easy to compute answers to questions like: How likely is John to call if there s an earthquake How likely is Mary to call on a random day? CIS Intro to AI 7
8 Semantics Local semantics give rise to global semantics Local semantics: given its parents, each node is conditionally independent of its other ancestors REVIEW (clarity is crucial): A and B are conditionally independent given C iff P(A B, C) = P(A C) Alternatively, P(B A, C) = P(B C) Alternatively, P(A B C) = P(A C) P(B C) Review: Toothache (T), Spot in Xray (X), Cavity (C) None of these propositions are independent of one other But T & X are conditionally independent given C T Effect 1 C Cause X Effect 2 CIS Intro to AI 8
9 Semantics Local semantics give rise to global semantics Local semantics: given its parents, each node is conditionally independent of its other ancestors Global semantics (why?): Generalizes Naïve Bayes P( J, M, A, B, E) P( B) P( E) P( A B, E) P( J A) P( M A) CIS Intro to AI 9
10 Naïve Bayes as a Bayesian Network Flu P(Flu).10 X 1 X 2 X 3 X 4 X 5 runnynose sinus cough fever muscle-ache Flu P(X 1 ) Flu P(X 2 ) Flu P(X 3 ) Flu P(X 4 ) Flu P(X 1 ) T.90 T.60 T.75 T.95 T.90 F.10 F.05 F.10 F.05 F.10 P ( 5 Flu, X1,, X 5) P( Flu) P( X1 Flu) P( X Flu) CIS Intro to AI 10
11 Belief Network Construction Algorithm 1. Choose some ordering of variables X 1,, X n 2. For i = 1 to n a. add X i to the network b. select parents from X 1,, X i-1 such that 3. Yields the smallest networks when causers (diseases) precede consequences (symptoms) CIS Intro to AI 11
12 Construction Example Suppose we choose the ordering M, J, A, B, E M J A P(J M)=P(J)? No P(A J,M)=P(A)? P(A J,M)=P(A J)? P(B A,J,M)=P(B)? P(B A,J,M)=P(B A)? No P(E B,A,J,M)=P(E A)? Yes P(E B,A,J,M)=P(E A,B)? B No Yes No E CIS Intro to AI 12
13 Notes: Belief Network Construction Details 1) order the variables (any ordering will do) A, B, C, D, E,.. 2) pop the first item off the list, and test to see what a minimal set of parents is i) add A (trivial) ii) add B test: is it true that P(B A) = P(B ~A) = P(B) if not, then add a link from A to B iii) add C test: is P(C) = P(C A,B) = P(C ~A,B)=P(C B,~A) = P(C ~A,~B) if so C is not linked to anything test: is P(C) = P(C A) = P(C ~A) if so then there is no link from A to C test: is P(C) = P(C B) = P(C ~B) if so then there is no link from B to C If some of the above tests fail, then there there will be links. One then has to test whether A alone, or B alone is sufficient? test: is P(C B) = P(C B,A) = P(C B, ~A) if so then we only need a link from B to C test: is P(C A) = P(C A,B) = P(C A, ~B) if so then we only need a link from A to C iv) add D things keep getting uglier, but the same idea continues. CIS Intro to AI 13
14 Lessons from the Example Network less compact: 13 numbers (compared to 10) Ordering of variables can make a big difference! Intuitions about causality useful CIS Intro to AI 14
15 How to build a belief net? Structure: Ask people: Try to model causality in structure Real causal structures yield the best networks Automatically: How to cast finding structure as search? Conditional Probability Tables Ask people: Get approximate probabilities from experts Automatically: Estimate probabilities from data CIS Intro to AI 15
16 Belief Net Extensions Hidden variables Decision (action) variables Continuous distributions Plate models E.g. Latent Dirichlet Allocation (LDA) Dynamical Belief Nets (DBNs) E.g. Hidden Markov Models (HMMs) CIS Intro to AI 16
17 Belief Networks and Hidden Variables: An Example CIS Intro to AI 17
18 Compact Conditional Probability Tables CPT size grows exponentially; continuous variables have infinite CPTs! Solution: compactly defined canonical distributions E.g., Boolean functions Gaussian and other standard distributions CIS Intro to AI 18
19 Summary Belief nets provide a compact representation of joint probability distributions Guaranteed to be a consistent specification Graphically show conditional independence Can be extended to include action nodes Networks that capture causality tend to be sparser Estimating belief nets is easy if everything is observable But requires serious search if the network structure is not known Not covered: Probabilistic inference algorithms given belief nets CIS Intro to AI 19
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From: AAAI Technical Report WS Compilation copyright 2000, AAAI (www.aaai.org). All rights reserved.
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