Here's a more serious situation where again the data we have contains information about behavior, but the thing we really want to infer is the hidden intent behind the behavior.

A government inspector specializing in securities fraud is looking at the stock trades in a particular company X made by a particular investor Y . Company X provides a detailed financial report and earnings announcement every quarter (that is, every three months), and the stock often moves after this announcement based on whether the company has reported good news or bad news.

There is no visible indication that investor Y is receiving any advance information about the contents of these quarterly reports and announcements, but the inspector has noticed a strange anomaly: for the past 12 quarters (totaling three years), each time the quarterly announcement contained good news, the investor bought a significant amount of company X's stock shortly beforehand (which benefitted them by acquiring stock that was about to rise in value); and each time the quarterly announcement contained bad news, the investor sold a significant amount of company X's stock shortly beforehand (which benefitted them by getting rid of stock that was about to fall in value).

It could be that this is all a coincidence, and investor Y has just been very lucky in their choices of when to buy and sell stock in company X over the past 12 quarters. But it could be that investor Y is able to complete this because they are receiving inside information that is being leaked to them by executives at the company before each quarterly announcement. (We will refer to this activity as insider trading.) The government inspector views it as important to decide which it is a coincidence or inside information since in the latter case, the investor might be violating federal laws against trading based on this type of inside information from within a company.

The inspector would like to get a rough estimate of the probability that investor Y 's trades in company X are based on inside information, given their pattern of trading over the past 12 quarters. To complete this, they start with some basic assumptions. From their experience, people with investor Y 's kinds of professional connections and in the absence of any other evidence have a 0.5% chance of being involved in insider trading with respect to any

given company. If they are involved in insider trading, then we can assume that they will buy stock just before each quarterly announcement containing good news, and sell stock just before each quarterly announcement containing bad news. If they are not involved in insider trading, then we can assume that the motion of the stock looks random to them: at any point in time, its probability of going up is 1/2 and its probability of going down is 1/2, independently of its behavior at any other point in time, and the investor can do no better than guessing with equal probability.

(3.1) (2 points) Suppose you want to express the probability that the inspector is looking for as the value of a conditional probability Pr[A | E], for some claim A and some evidence E. What would you choose for A and E so that Pr[A | E] is the value we're seeking? Give an explanation for your answer.

(3.2) (5 points) Apply Bayes' Rule to your choice of A and E from (3.1), to compute Pr[A | E], the probability that the inspector is trying to work out. As part of this, show the calculations you use to start from the formula for Bayes' Rule and to arrive at the value of conditional probability.

Before proceeding further, the government inspector checks with a data scientist who has experience studying company X and its financial performance over time. The data scientist says that in fact the behavior of company X's stock is relatively predictable around the time of its quarterly announcements, because there is so much relevant public information being reported about the company during these periods. The data scientists's analysis suggests that if an investor were to study all these public information which they could do completely legally without ever having access to inside information they could correctly predict whether the stock will move up or down after the quarterly announcement with probability 0.8.

So if we go back to the government inspector's original assumption that if investor Y is not using inside information then they can only predict the direction of company X's stock with probability 1/2 we see that it seems too harsh in retrospect. A more reasonable assumption, given what the data scientist has reported, is that if investor Y is not using inside information then they can correctly predict the direction of company X's stock before a quarterly announement with probability 0.8, independently of whether they are correct at any other point in time.

(3.3) (4 points) This revised assumption changes the value of Pr [A | E], the probability that the inspector is trying to work out. Apply Bayes' Rule to compute the new value of Pr[A | E] based on the new assumption. As part of this, show the calculations you use to start from the formula for Bayes' Rule and to arrive at the value of conditional probability.

(3.4) (2 points) In both (3.2) and (3.3), you computed Pr[A | E], the probability the in- spector was interested in, based on different assumptions about how investor Y guesses the direction of price change for company X's stock. Which computation gave the smaller value

for Pr[A | E]? Give an informal explanation without going into any further calculations for why this was the smaller one.