Problem Set 5

Assigned: Apr. 5
Due: Apr. 19

Problem 1

Consider the following Bayesian network:

Let A and B be Boolean variables, C have values 1 and 2, and D have values 1, 3, and 10. The following probabilities are given
 P(A=T) = 0.9 P(A=F) = 0.1 P(B=T) = 0.4 P(B=F) = 0.6 P(C=1|A=T,B=T) = 0.9 P(C=2|A=T,B=T) = 0.1 P(C=1|A=T,B=F) = 0.6 P(C=2|A=T,B=F) = 0.4 P(C=1|A=F,B=T) = 0.3 P(C=2|A=F,B=T) = 0.7 P(C=1|A=F,B=F) = 0.0 P(C=2|A=F,B=F) = 1.0 P(D=1|C=1) = 0.7 P(D=3|C=1) = 0.2 P(D=10|C=1) = 0.1 P(D=2|C=2) = 0.2 P(D=3|C=2) = 0.3 P(D=10|C=2) = 0.5
Compute the following
• P(C=1|A=T).
• P(A=T|C=1).
• P(B=T|A=T).
• The (unconditional) probability distribution for D.
• E(C) (The expected value)
• E(D)
• E(C|A=T)
• E(C+D)
• E(C*D)

Problem 2

A publisher needs to decide whether to publish a book that has been submitted. Let us simplify the situation by assuming that, if the book is published, there are two possible outcomes: Success and Failure. If Success then the publisher will make \$50,000; if Failure the publisher will lose \$10,000. If the publisher doesn't publish the book, then he breaks even. The publisher can hire a reviewer for \$500. Again, we will assume that the reviewer will give one of two possible reviews: For or Against. We are given the following probabilities

 Prob(Success) = 0.2 Prob(Failure)=0.8. Prob(For|Success) = 0.7 Prob(Against|Success) = 0.3 Prob(For|Failure) = 0.4 Prob(Against|Failure) = 0.6

A. What is the publisher's expected gain from consulting the reviewer?

B. The publisher also has the option of consulting with two reviewers. Assume that the two reviewers follow the same probabilistic model, and that their reviews are conditionally independent given the actual success or failure. What is the publisher's optimal strategy; in particular, if he gets one favorable and one unfavorable review, what should his decision be? What is the expected gain from consulting with two reviewers?

C. (Extra credit) What is the optimal number of reviewers to consult with, and what is the optimal strategy? Suppose that a reviewer only costs \$100; what then? Feel free to program this rather than work it out by hand.