Question $1$: Bayesian Networks, Netica $(30+10+5+5=50$ marks$)$

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, whereas only $0\mathrm{.}4\%$ of the transactions are fraudulent when they are 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, whereas 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$$t own any computer. Currently, $70\%$ of the population owns a computer or smart phone, and for those card holders, $1\%$ of their legitimate transactions are done over the Internet, but this percentage increases to $2\%$ for fraudulent transactions. For those who don$$t own any computer or smart phone, a mere $0\mathrm{.}1\%$ of their legitimate transactions is done over the Internet, but that number increases to $1\mathrm{.}1\%$ for fraudulent transactions. Unfortunately, the credit card company doesn$$t know whether a card holder owns a computer or smart phone, but it can usually guess by verifying whether any of the recent transactions involve the purchase of computer related accessories. In any given week, $10\%$ of those who own a computer or smart phone purchase $($with their credit card$)$ at least one computer related item, as opposed to just $0\mathrm{.}1\%$ of those who don$$t own any computer or smart phone.

$($a$)$ Construct a Bayesian Network $($BN$)$ to identify fraudulent transactions. This network

should encode the information stated above. Your network should contain exactly six

nodes, corresponding to the following binary random variables:

$(30$ marks$)$

$$ OC $$ 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 Bayes Network should accurately capture the probabilistic depen$-$

dencies between these variables.

What to hand in:

$$ The graph defining the network and the Conditional Probability Tables $($CPTs$)$ asso$-$

ciated with each node in the graph.

$$ A Netica file of the BN$,$ and a screenshot of your BN and the CPTs for the nodes.

$($b$)$ Consider the following questions.

i$.$ What is the prior probability $($before we search for previous computer related purchases

and before we verify whether it is a foreign and$/$or an Internet purchase$)$ that the current

transaction is a fraud?

$(1$ mark$)$

ii$.$ 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? Enter each piece of evi$-$dence in sequence and update the probability of interest in turn. Use the structure of the BN and the CPTs to explain the impact of each new piece of evidence. $(9$ marks$)$ What to hand in:

$$ Explanations of how these pieces of evidence affect the probabilities of interest in the

BN$.$

$$ Screenshots of the changes to the BN after each piece of evidence is added $($one

screenshot per piece of evidence$).$

$($c$)$ After computing the probabilities in item $($b$)$ii$,$ the fraud detection system raises a flag and

recommends that the card holder be called to confirm the transaction. An agent calls at the domicile of the card holder, but they are not home. Their spouse confirms that they are currently out of town on a business trip. How does the probability of a fraud obtained in item $($b$)$ii change based on this new piece of information? What happens if you remove

the evidence about the foreign purchase? Why?

$(5$ marks$)$

What to hand in:

$$ An explanation of how this information affects the probability of fraud, and any other

relevant probabilities in the BN$.$

$$ A screenshot of the change to the BN after the new evidence is added.

$($d$)$ Suppose you are not a very honest employee, and you just stole a credit card. You know

that the fraud detection system uses the BN designed earlier, but you still want to make an important purchase over the Internet. Keeping in mind what the credit card company only has access to the records of your purchases, what can you do to reduce the risk that the credit card company will deem the transaction to