CSE 473 - Artificial Intelligence - Autumn 2011

Written Assignment 3: Probability and Bayesian Networks

Due Date: Friday Dec 9, 9:30am in class or through the online dropbox.

Problem 1: Simple Spam [15 points]

According to extremely reliable sources (Wikipedia), 78% of email is spam. According to experiments conducted on my own inbox, 11% of spam email messages contain the word "Pills". In comparison, only 1% of non-spam email messages contain this word.

A. [5pts] What is the probability that a message contains the word "Pills" and is Spam?

B. [5pts] What is the probability that a message is Spam if it is known to contain the word "Pills"?

C. [5pts] What is the probability that a message does not contain the word "Pills" or is Spam?

Problem 2: Bayesian Modeling [20 points]

Consider the following modeling challenge, which might arise when working on a homework problem. You can either study hard or trick the TA into telling you the right answer. Given these choices, you might actually learn the material and you might also get a passing grade on the problem. All of these possibilities will happen with certain probabilities, that you are welcome to imagine however you like.

A. [14pts] Model this problem as a Bayesian network representing a joint distribution over four binary random variables. Since there is more than one possible answer, briefly motivate your choices.

B. [6pts] Write three independence assumptions that your network encodes.

Problem 3: Bayesian Politics [40 points]

Consider the above Bayes net.

The variable are boolean and describe aspects of a court trail. They indicate whether someone broke an election law (B), was indicted (I), whether the prosecutor was politically motivated (M), if the person was found guilty (G), and if they were ultimately put in jail (J).

A. [6pts] Which of the following are true:

B. [4pts] What is P(b,i,¬m,g,j)?

C. [20pts] Use variable elimination to calculate the probability someone goes to jail, given that he or she broke the law and the prosecutor was not politically motivated. Try to keep the intermediate factors as small as possible. Describe all of the assumptions/decisions that you make at each step.

D. [10pts] Describe the process by which you might sample a joint assignment to the variables of this Bayes net, conditioned on the fact that there was a guilty verdict. You can use any of the techniques we described in class, but you should justify your choice.