Problem Set #3

Due:  March 1, 2012 by the start of class

Formatting and Submission Instructions

Your responses to these questions must be typed and submitted in hardcopy.  For this assignment, you may check your answers with another student and work through the problems together only AFTER you have made a serious attempt by yourself.  Make absolutely certain that you can do these types of problems by yourself, since they will be on the exams.

Be certain to include a statement of sources at the top of your assignment, listing all sources you consulted (websites, fellow students, etc.) while completing the assignment.  You do not need to list any course materials (textbooks, lecture notes, or the professor).

Bayesian Networks and Probability

An AI student notices that people who drive SUVs (S) consume large amounts of gas (G) and are involved in more accidents (A) than the national average.  In this problem, uppercase letters denote the variable names, lowercase values denote the variables with value. For example, P(a, ~s) is the same thing as P(A = t, S = f). She has constructed the following Bayesian network:

1.) (20 pts.) Compute P(a, ~s, g) using the chain rule.

2.) (20 pts.) Compute P(a) using inference by enumeration.

3.) (20 pts.) Using conditional independence, compute P(~g, a | s) and P(~g, a | ~s).  Then use Bayes' rule to compute P(s | ~g, a).

The enterprising student notices that two types of people drive SUVs: people from California (C) and people with large families (F). After collecting some statistics, the student arrives at the following form for the Bayes Net:

4.) (20 pts.) Using the chain rule, compute the probability P(~g, a, s, c, ~f).

5.) (20 pts) Using the rules for determining when two variables are (conditionally) independent of each other in a Bayes Net, answer the following (True/False) for the Bayes Net given above:

1. Independent(C, G)
2. Independent(F, A | S)
3. Independent(C, F)
4. Independent(A, G)
5. Independent(C, F | A)