Question

4. Conditional Independencies in Bayes Nets: A E M A A M (ii) (iii) (i) The Bayesian networks in Figure above are all part of the alarm network introduced in class and in Russell and Norvig. We use the notation XIY to denote the variable X being independent of Y, and XIY |Z to denote X being independent of Y given Z d. Similarly, prove that BLM |A in the Bayesian network of Fig. (ii), and MIJ|A in the Bayesian network of Fig. (ii)

Answer 1

d)

P(B)P(A|B) = P(A)P(B|A) from Bayes rule.

Therefore P(B, A, M) = P(B)P(A|B)P(M|A)

= P(A)P(B|A)P(M|A).

P(B, M|A) = P(B, A, M)/P(A)

= P(A)P(B|A)P(M|A)/P(A)

= P(B|A)P(M|A)

Therefore B⊥M|A. P(M, A, J) = P(A)P(M|A)P(J|A).

Therefore P(M, J|A) = P(M, J, A)/P(A)

= P(A)P(M|A)P(J|A)/P(A)

= P(M|A)P(J|A).

Therefore M⊥J|A