CS440 - INTRODUCTION TO ARTIFICIAL INTELLIGENCE  - Assignment 3 - Solved

Adversarial Search - Bayesian Networks


Problem 1 (10 points) : Consider the two-player game described in Figure 1.

1.   Draw the complete game tree, using the following conventions:

2.   Write each state as (sA,sB) where sA and sB denote the token locations.

3.   Put each terminal state in a square box and write its game value in a circle.

4.   Put loop states (states that already appear on the path to the root) in double square boxes. Since it is not clear how to assign values to loop states, annotate each with a “?” in a circle.

5.   Now mark each node with its backed-up minimax value (also in circle). Explain how you handled the “?” values andwhy.

 

Figure 1: The start position of a simple game. Player A moves first. The two players take turns moving, and each player must move his token to an open adjacent space in either direction. If the opponent occupies an adjacent space, than a player may jump over the opponent to the next open space if any. (For example, if A is on 3 and B is on 2, then A may move back to 1.) The game ends when one player reaches the opposite end of the board. If player A reaches space 4 first, then the value of the game to A is +1; if player B reaches space 1 first, then the value of the game to A is -1.

Problem 2 (5 points): Consider the following Bayesian network, where variables A through E are all Boolean valued. Note: there is a typo in the image, it should be P(A = true) = 0.2 instead of P(D = true) = 0.2 .

 

a)    What is the probability that all five of these Boolean variables are simultaneously true?

[Hint: You have to compute the joint probability distribution. The structure of the Bayesian network suggests how the joint probability distribution is decomposed to the conditional probabilities available.]

b)   What is the probability that all five of these Boolean variables are simultaneously false?[Hint: Answer similarly to above.]

c)    What is the probability that A is false given that the four other variables are all known to be true?

Problem 3 (5 points):

a)    Calculate P(Burglary|JohnsCalls = true,MaryCalls = true) and show in detail the calculations that take place. Use your book to confirm that your answer is correct.

b)   Suppose a Bayesian network has the from of a chain: a sequence of Boolean variables X1,...Xn where Parents(Xi) = {Xi−1} for i = 2,...,n. What is the complexity of computing P(X1|Xn = true) using enumeration? What is the complexity with variable elimination?

 

Problem 4 (10 points): Suppose you are working for a financial institution and you are asked to implement a fraud detection system. You plan to use the following information:

•    When the card holder is travelling abroad, fraudulent transactions are more likely since tourists are prime targets for thieves. More precisely, 1% of transactions are fraudulent when the card holder is travelling, where as only 0.4% of the transactions are fraudulent when she is not travelling. On average, 5% of all transactions happen while the card holder is travelling. If a transaction is fraudulent, then the likelihood of a foreign purchase increases, unless the card holder happens to be travelling. More precisely, when the card holder is not travelling, 10% of the fraudulent transactions are foreign purchases where as only 1% of the legitimate transactions are foreign purchases. On the other hand, when the card holder is travelling, then 90% of the transactions are foreign purchases regardless of the legitimacy of the transactions.

•    Purchases made over the internet are more likely to be fraudulent. This is especially true for card holders who don’t own any computer. Currently, 75% of the population owns a computer or smart phone and for those card holders, 1% of their legitimate transactions are done over the internet, however this percentage increases to 2% for fraudulent transactions. For those who don’t own any computer or smart phone, a mere 0.1% of their legitimate transactions is done over the internet, but that number increases to 1.1% for fraudulent transactions. Unfortunately, the credit card company doesn’t know whether a card holder owns a computer or smart phone, however it can usually guess by verifying whether any of the recent transactions involve the purchase of computer related accessories. In any given week, 10% of those who own a computer or smart phone purchase (with their credit card) at least one computer related item as opposed to just 0.1% of those who don’t own any computer or smart phone.

a)    Construct a Bayes Network to identify fraudulent transactions.


OC : card holder owns a computer or smart phone.

Fraud : current transaction is fraudulent.

Trav : card holder is currently travelling.

FP : current transaction is a foreign purchase.

IP : current purchase is an internet purchase.

CRP : a computer related purchase was made in the past week.

The arcs defining your Network should accurately capture the probabilistic dependencies between these variables.

b)   What is the prior probability (i.e., before we search for previous computer related purchases and before we verifywhether it is a foreign and/or an internet purchase) that the current transaction is a fraud? What is the probability that the current transaction is a fraud once we have verified that it is a foreign transaction, but not an internet purchase and that the card holder purchased computer related accessories in the past week?