Bayesian Networks_Assignment 8 Solution

CS161: FUNDAMENTALS OF ARTIFICIAL INTELLIGENCE
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1. Prove:
Generalized product rule: Pr(A,B | K) = Pr(A | B,K)Pr(B | K). Generalized Bayes’ rule: Pr(A | B,K) = Pr(B | A,K)Pr(A | K)/Pr(B | K).
2. An oil well may be drilled on Mr. Y’s farm in Texas. Based on what has happened to similar farms, we judge the probability of oil being present to be .5, the probability of only natural gas being present to be .2, and the probability of neither being present to be .3. If oil is present, a geological test will give a positive result with probability .9; if only natural gas is present, it will give a positive result with probability .3; and if neither are present, the test will be positive with probability .1. Suppose the test comes back positive. What’s the probability that oil is present?
3. (From Bayesian Networks by R. Neapolitan) Consider the set of objects below.
1 1 2 2 2 2
1 2
1 2 2
1 2
Mr. Y picked up an object at random from the above set. We want to compute the probabilities of the following events:
α1: the object is black; α2: the object is square; α3: if the object is one or black, then it is also square.
1
A B
C D E
F
H
G
Figure 1: The DAG of a Bayesian network.
Construct the joint probability distribution of this problem. Use it to compute the above probabilities by explicitly identifying the worlds at which each αi holds. Identify two sets of sentences α,β,γ such that α is independent of β given γ with respect to the constructed distribution.
4. Consider the DAG in Figure 1:
(a) List the Markovian assumptions asserted by the DAG. (b) True or false? Why? • d separated(A,BH,E) • d separated(G,D,E) • d separated(AB,F,GH) (c) Express Pr(a,b,c,d,e,f,g,h) in factored form using the chain rule for Bayesian networks. (d) Compute Pr(A = 0,B = 0) and Pr(E = 1 | A = 1). Justify your answers.
Pr(A = 0) Pr(A = 1) .8 .2
Pr(B = 0) Pr(B = 1) .3 .7
Pr(E = 0 | B) Pr(E = 1 | B) B = 0 .1 .9 B = 1 .9 .1
Pr(D = 0 | A,B) Pr(D = 1 | A,B) A = 0,B = 0 .2 .8 A = 0,B = 1 .9 .1 A = 1,B = 0 .4 .6 A = 1,B = 1 .5 .5