1. Bayesian networks Chapter 14 Slide Set 2

2. Constructing Bayesian networks 1. Choose an ordering of variables X 1, …,X n 2. For i = 1 to n –add X i to the network –select parents from X 1, …,X i-1 such that P (X i | Parents(X i )) = P (X i | X 1,... X i-1 )

3. Suppose we choose the ordering M, J, A, B, E P(J | M) = P(J)? Example

4. Suppose we choose the ordering M, J, A, B, E P(J | M) = P(J)? No P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? Example

5. Suppose we choose the ordering M, J, A, B, E P(J | M) = P(J)? No P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? No P(B | A, J, M) = P(B | A)? P(B | A, J, M) = P(B)? Example

6. Suppose we choose the ordering M, J, A, B, E P(J | M) = P(J)? No P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? No P(B | A, J, M) = P(B | A)? Yes P(B | A, J, M) = P(B)? No P(E | B, A,J, M) = P(E | A)? P(E | B, A, J, M) = P(E | A, B)? Example

7. Suppose we choose the ordering M, J, A, B, E P(J | M) = P(J)? No P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? No P(B | A, J, M) = P(B | A)? Yes P(B | A, J, M) = P(B)? No P(E | B, A,J, M) = P(E | A)? No P(E | B, A, J, M) = P(E | A, B)? Yes Example

8. Example contd. Deciding conditional independence is hard in noncausal directions (Causal models and conditional independence seem hardwired for humans!) Network is less compact: 1 + 2 + 4 + 2 + 4 = 13 numbers needed

9. 9 Using a Bayesian Network Suppose you want to calculate: P(A = true, B = true, C = true, D = true) = P(A = true) * P(B = true | A = true) * P(C = true | B = true) P( D = true | B = true) = (0.4)*(0.3)*(0.1)*(0.95) A B CD

10. 10 Using a Bayesian Network Example Using the network in the example, suppose you want to calculate: P(A = true, B = true, C = true, D = true) = P(A = true) * P(B = true | A = true) * P(C = true | B = true) P( D = true | B = true) = (0.4)*(0.3)*(0.1)*(0.95) A B CD This is from the graph structure These numbers are from the conditional probability tables

11. 11 Inference Using a Bayesian network to compute probabilities is called inference In general, inference involves queries of the form: P( X | E ) X = The query variable(s) E = The evidence variable(s)

12. 12 Inference An example of a query would be: P( HasAnthrax = true | HasFever = true, HasCough = true) Note: Even though HasDifficultyBreathing and HasWideMediastinum are in the Bayesian network, they are not given values in the query (ie. they do not appear either as query variables or evidence variables) They are treated as unobserved (hidden) variables HasAnthrax HasCoughHasFeverHasDifficultyBreathingHasWideMediastinum

13. 13 The Bad News Exact inference in BBNs is NP-hard –Though feasible for singly-connected networks –But we can achieve significant improvements (e.g., variable elimination) There are approximate inference techniques which are much faster and give fairly good results Next class: inference

14. Example In your local nuclear power plant, there is an alarm that senses when a temperature gauge exceeds a given threshold. The gauge measures the temperature of the core. Consider the Boolean variables: A (alarm sounds), FA (alarm faulty), FG (gauge is faulty), and the multivalued variables G (gauge reading) and T (actual core temperature. The gauge is more likely to fail when the core temperature gets too high Let’s draw the network (in class)

15. Example (cont) Suppose –there are just two possible actual and measured temperatures, normal and high –The prob that the gauge gives the correct temp is X when it is working, but Y when it is faulty. Give the CPT for G (in class) Suppose –The alarm works correctly unless it is faulty, in which case it never sounds. Give the CPT for A (in class)

16. Example (cont.) Suppose FA=false; FG=false (the alarm and gauge are working properly); and A=True (and the alarm sounds). What is the probability that the temperature is too high? (T=high?) (in class)