## Problem Set 6

Assigned: Nov. 1
Due: Nov. 8

Suppose that we are given the following probabilistic model relating the likelihood that two people working in the same field at the same time ever met to certain other random variables:

Random Variables:

M: Did the two individuals ever meet, and what was the outcome?
Three possible values: 1=met and liked; 2=met and disliked; 3=never met.

C: Did the two individuals correspond?
Values: T or F.

R: Did the two publish reviews/criticisms of one another's work?
Values: 1=positive reviews; 2=negative reviews; 3=no reviews.

A: How close were the people in age?
Values: 1=less than 10 years apart; 2=at least 10 years apart.

D: How near one another did the people live?
Values: 1=same country; 2=different countries.

There is a Bayesian network with the following structure:

The following conditional probabilities are recorded:

 P(A=1)=0.2 P(A=2) = 0.8.

 P(D=1)=0.1 P(D=2) = 0.9.

 P(M=1|A=1,D=1) = 0.6 P(M=2|A=1,D=1) = 0.4 P(M=3|A=1,D=1) = 0 P(M=1|A=1,D=2) = 0.4 P(M=2|A=1,D=2) = 0.2 P(M=3|A=1,D=2) = 0.5 P(M=1|A=2,D=1) = 0.5 P(M=2|A=2,D=1) = 0.2 P(M=3|A=2,D=1) = 0.3 P(M=1|A=2,D=2) = 0.3 P(M=2|A=2,D=2) = 0.1 P(M=3|A=2,D=2) = 0.6

 P(C=T|M=1) = 0.8 P(C=F|M=1) = 0.2 P(C=T|M=2) = 0.1 P(C=F|M=2) = 0.9 P(C=T|M=3) = 0.4 P(C=F|M=3) = 0.6

```

```

 P(R=1|M=1) = 0.5 P(R=2|M=1) = 0.1 P(R=3|M=1) = 0.4 P(R=1|M=2) = 0.2 P(R=2|M=2) = 0.2 P(R=3|M=2) = 0.6 P(R=1|M=3) = 0.2 P(R=2|M=3) = 0.1 P(R=3|M=3) = 0.7

Not to hand in, but you might want to think about in preparation for the final essay:
Obviously (a) this model is structurally absurd; (b) I made up the numbers off the top of my head. Questions:

• Can you be more specific about what is wrong with the structure of the model?
• How would you go about designing a better structural model?
• How would you go about assigning the conditional probabilities that you would need for your model?
Note: These questions are not at all easy.

To hand in:

Evaluate the following probabilities:

• 1. P(M=1).
• 2. P(M=1|A=1).
• 3. P(A=1|M=1).
• 4. P(M=1|C=N)
• 5. P(M=1|A=1,C=N)
• 6. P(R=1|C=Y)