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- 1. Bayesian Networks CSC 371: Spring 2012
- 2. Today’s Lecture• Recap: Joint distribution, independence, marginal independence, conditional independence• Bayesian networks• Reading: – Sections 14.1-14.4 in AIMA [Russel & Norvig]
- 3. Marginal Independence• Intuitively: if X ╨ Y, then – learning that Y=y does not change your belief in X – and this is true for all values y that Y could take• For example, weather is marginally independent of the result of a coin toss3
- 4. Marginal Independence4
- 5. Conditional Independence• Intuitively: if X ╨ Y | Z, then – learning that Y=y does not change your belief in X when we already know Z=z – and this is true for all values y that Y could take and all values z that Z could take• For example, 5 ExamGrade ╨ AssignmentGrade | UnderstoodMaterial
- 6. Conditional Independence
- 7. “…probability theory is more fundamentally concerned withthe structure of reasoning and causation than withnumbers.” Glenn Shafer and Judea Pearl Introduction to Readings in Uncertain Reasoning, Morgan Kaufmann, 1990
- 8. Bayesian Network Motivation• We want a representation and reasoning system that is based on conditional (and marginal) independence – Compact yet expressive representation – Efficient reasoning procedures• Bayesian (Belief) Networks are such a representation – Named after Thomas Bayes (ca. 1702 –1761) – Term coined in 1985 by Judea Pearl (1936 – ) – Their invention changed the primary focus of AI from logic to probability! Thomas Bayes Judea Pearl 8
- 9. Bayesian Networks: Intuition• A graphical representation for a joint probability distribution – Nodes are random variables • Can be assigned (observed) or unassigned (unobserved) – Arcs are interactions between nodes • Encode conditional independence • An arrow from one variable to another indicates direct influence • Directed arcs between nodes reflect dependence – A compact specification of full joint distributions• Some informal examples: Smoking At Fire Understood Sensor Material Assignment Exam Grade Grade Alarm9
- 10. Example of a simple Bayesian network A B p(A,B,C) = p(A)p(B) p(C|A,B) C• Probability model has simple factored form• Directed edges => direct dependence• Absence of an edge => conditional independence• Also known as belief networks, graphical models, causal networks• Other formulations, e.g., undirected graphical models
- 11. Bayesian Networks: Definition11
- 12. Bayesian Networks: Definition• Discrete Bayesian networks: – Domain of each variable is finite – Conditional probability distribution is a conditional probability table – We will assume this discrete case • But everything we say about independence (marginal & conditional)12 carries over to the continuous case
- 13. Examples of 3-way Bayesian Networks A B C Marginal Independence: p(A,B,C) = p(A) p(B) p(C)
- 14. Examples of 3-way Bayesian Networks Conditionally independent effects: p(A,B,C) = p(B|A)p(C|A)p(A) A B and C are conditionally independent Given A e.g., A is a disease, and we modelB C B and C as conditionally independent symptoms given A
- 15. Examples of 3-way Bayesian Networks A B Independent Causes: p(A,B,C) = p(C|A,B)p(A)p(B) C “Explaining away” effect: Given C, observing A makes B less likely e.g., earthquake/burglary/alarm example A and B are (marginally) independent but become dependent once C is known
- 16. Examples of 3-way Bayesian Networks A B C Markov dependence: p(A,B,C) = p(C|B) p(B|A)p(A)
- 17. Example: Burglar Alarm• I have a burglar alarm that is sometimes set off by minor earthquakes. My two neighbors, John and Mary, promised to call me at work if they hear the alarm – Example inference task: suppose Mary calls and John doesn’t call. What is the probability of a burglary?• What are the random variables? – Burglary, Earthquake, Alarm, JohnCalls, MaryCalls
- 18. Example 5 binary variables: B = a burglary occurs at your house E = an earthquake occurs at your house A = the alarm goes off J = John calls to report the alarm M = Mary calls to report the alarm What is P(B | M, J) ? (for example) We can use the full joint distribution to answer this question Requires 25 = 32 probabilities Can we use prior domain knowledge to come up with a Bayesian network that requires fewer probabilities? What are the direct influence relationships? A burglary can set the alarm off An earthquake can set the alarm off The alarm can cause Mary to call The alarm can cause John to call
- 19. Example: Burglar Alarm What are the model parameters?
- 20. Conditional probability distributions• To specify the full joint distribution, we need to specify a conditional distribution for each node given its parents: P (X | Parents(X)) Z1 Z2 … Zn X P (X | Z1, …, Zn)
- 21. Example: Burglar Alarm
- 22. The joint probability distribution• For each node Xi, we know P(Xi | Parents(Xi))• How do we get the full joint distribution P(X1, …, Xn)?• Using chain rule: n n P ( X 1 , , X n ) = ∏ P( X i | X 1 , , X i −1 ) = ∏ P( X i | Parents( X i ) ) i =1 i =1• For example, P(j, m, a, ¬b, ¬e) = P(¬b) P(¬e) P(a | ¬b, ¬e) P(j | a) P(m | a)•
- 23. Constructing a Bayesian Network: Step 1 • Order the variables in terms of causality (may be a partial order) e.g., {E, B} -> {A} -> {J, M} • P(J, M, A, E, B) = P(J, M | A, E, B) P(A| E, B) P(E, B) ~ P(J, M | A) P(A| E, B) P(E) P(B) ~ P(J | A) P(M | A) P(A| E, B) P(E) P(B) These CI assumptions are reflected in the graph structure of the Bayesian network
- 24. Constructing this Bayesian Network: Step 2• P(J, M, A, E, B) = P(J | A) P(M | A) P(A | E, B) P(E) P(B)• There are 3 conditional probability tables (CPDs) to be determined: P(J | A), P(M | A), P(A | E, B) – Requiring 2 + 2 + 4 = 8 probabilities• And 2 marginal probabilities P(E), P(B) -> 2 more probabilities • 2 + 2 + 4 + 1 + 1 = 10 numbers (vs. 25-1 = 31)• Where do these probabilities come from? – Expert knowledge – From data (relative frequency estimates) – Or a combination of both
- 25. Number of Probabilities in Bayesian Networks• Consider n binary variables• Unconstrained joint distribution requires O(2n) probabilities• If we have a Bayesian network, with a maximum of k parents for any node, then we need O(n 2k) probabilities• Example – Full unconstrained joint distribution • n = 30: need 109 probabilities for full joint distribution – Bayesian network • n = 30, k = 4: need 480 probabilities
- 26. Constructing Bayesian networks1. Choose an ordering of variables X1, … , Xn2. For i = 1 to n – add Xi to the network – select parents from X1, … ,Xi-1 such that P(Xi | Parents(Xi)) = P(Xi | X1, ... Xi-1)
- 27. Example• Suppose we choose the ordering M, J, A, B, EP(J | M) = P(J)?
- 28. Example• Suppose we choose the ordering M, J, A, B, EP(J | M) = P(J)? No
- 29. Example• Suppose we choose the ordering M, J, A, B, EP(J | M) = P(J)? NoP(A | J, M) = P(A)?P(A | J, M) = P(A | J)?P(A | J, M) = P(A | M)?
- 30. Example• Suppose we choose the ordering M, J, A, B, EP(J | M) = P(J)? NoP(A | J, M) = P(A)? NoP(A | J, M) = P(A | J)? NoP(A | J, M) = P(A | M)? No
- 31. Example• Suppose we choose the ordering M, J, A, B, EP(J | M) = P(J)? NoP(A | J, M) = P(A)? NoP(A | J, M) = P(A | J)? NoP(A | J, M) = P(A | M)? NoP(B | A, J, M) = P(B)?P(B | A, J, M) = P(B | A)?
- 32. Example• Suppose we choose the ordering M, J, A, B, EP(J | M) = P(J)? NoP(A | J, M) = P(A)? NoP(A | J, M) = P(A | J)? NoP(A | J, M) = P(A | M)? NoP(B | A, J, M) = P(B)? NoP(B | A, J, M) = P(B | A)? Yes
- 33. Example• Suppose we choose the ordering M, J, A, B, EP(J | M) = P(J)? NoP(A | J, M) = P(A)? NoP(A | J, M) = P(A | J)? NoP(A | J, M) = P(A | M)? NoP(B | A, J, M) = P(B)? NoP(B | A, J, M) = P(B | A)? YesP(E | B, A ,J, M) = P(E)?P(E | B, A, J, M) = P(E | A, B)?
- 34. Example• Suppose we choose the ordering M, J, A, B, EP(J | M) = P(J)? NoP(A | J, M) = P(A)? NoP(A | J, M) = P(A | J)? NoP(A | J, M) = P(A | M)? NoP(B | A, J, M) = P(B)? NoP(B | A, J, M) = P(B | A)? YesP(E | B, A ,J, M) = P(E)? NoP(E | B, A, J, M) = P(E | A, B)?Yes
- 35. Example contd.• Deciding conditional independence is hard in noncausal directions – The causal direction seems much more natural• Network is less compact: 1 + 2 + 4 + 2 + 4 = 13 numbers needed
- 36. A more realistic Bayes Network: Car diagnosis• Initial observation: car won’t start• Orange: “broken, so fix it” nodes• Green: testable evidence• Gray: “hidden variables” to ensure sparse structure, reduce parameteres
- 37. The Bayesian Network from a different Variable Ordering
- 38. Given a graph, can we “read off” conditional independencies?
- 39. Are there wrong network structures?• Some variable orderings yield more compact, some less compact structures – Compact ones are better – But all representations resulting from this process are correct – One extreme: the fully connected network is always correct but rarely the best choice• How can a network structure be wrong? – If it misses directed edges that are required
- 40. Summary• Bayesian networks provide a natural representation for (causally induced) conditional independence• Topology + conditional probability tables• Generally easy for domain experts to construct
- 41. Probabilistic inference• A general scenario: – Query variables: X – Evidence (observed) variables: E = e – Unobserved variables: Y• If we know the full joint distribution P(X , E, Y), how can we perform inference about X? P( X , e) P( X | E = e) = ∝ ∑ y P( X , e, y ) P (e )• Problems – Full joint distributions are too large – Marginalizing out Y may involve too many summation terms
- 42. Conclusions…• Full joint distributions are intractable to work with – Conditional independence assumptions allow us to model real- world phenomena with much simpler models – Bayesian networks are a systematic way to construct parsimonious structured distributions• How do we do inference (reasoning) in Bayesian networks? – Systematic algorithms exist – Complexity depends on the structure of the graph

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