Figure 1: A Bayesian network graph establishing relations between various car problems and their causes.
On the network shown in Figure 1, consider the event E = "battery age is less than three years". Let A = P(E | "battery dead"=true AND "no oil"=true) and B = P(E | "battery dead"=true AND "no oil"=false). Which of the following three cases can possibly be true: A > B, A = B, or A < B? Why?
On the network shown in Figure 1, consider again the event E = "battery age is less than three years". Let A = P(E | "battery dead"=true AND "no oil"=true) and B = P(E | "battery dead"=true AND "starter broken"=true). Which of the following three cases can possibly be true: A > B, A = B, or A < B? Why?
The "alternator broken" event and the "fanbelt broken" event are both causes of the "no charging event." Let A = P("alternator broken"=true | "no charging"=true) and B = P("alternator broken"=true | "no charging"=true AND "fanbelt broken"=true). Which of the following three cases do you expect to be true: A > B, A = B, or A < B? Why?
On the network shown in Figure 1, the "no gas" event is a cause of the "car won't start" event. Let A = P("no gas"=true) and B = P("no gas"=true | "car won't start"=true). Which of the following three cases do you expect to be true: A > B, A = B, or A < B? Why?
On the network shown in Figure 1, suppose that:
Suppose that:
Figure 2: A Bayesian network establishing relations between events on the burglary-earthquake-alarm domain, together with complete specifications of all probability distributions.
On the network shown in Figure 2, what is the probability of the following event: burglary=false AND earthquake=true AND alarm=false AND JohnCalls=true AND MaryCalls=false.
On the network shown in Figure 2, what is the probability of the following event: earthquake=true AND alarm=false AND JohnCalls=true AND MaryCalls=false.
Figure 3: A decision tree for estimating whether the patron will be willing to wait for a table at a restaurant.
Suppose that, on the entire set of training samples available for constructing the decision tree of Figure 3, 80 people decide to wait, and 20 people decide not to wait. What is the initial entropy at node A (before the test is applied)? Write an expression that fully specifies the answer numerically.
In the decision tree of Figure 3, node D uses the exact same test (whether it is weekend or not) as node A. What is the information gain, at node D, of using the weekend test? Justify your answer.
Your boss at a software company gives you a binary classifier H that predicts, for any basketball game, whether the home team will win or not. Classifier H has a 28% accuracy, and your boss assigns you the task of improving that classifier, so that you get an accuracy that is better than 60%. How do you achieve that task?
You are a meteorologist that places temperature sensors all of the world, and you set them up so that they automatically e-mail you, each day, the high temperature for that day. Unfortunately, you have forgotten whether you placed a certain sensor S in Maine or in the Sahara desert (but you are sure you placed it in one of those two places) . The probability that you placed sensor S in Maine is 5%. The probability of getting a daily high temperature of 80 degrees or more is 20% in Maine and 90% in Sahara. Assume that probability of a daily high for any day is independent of the daily high for the previous day.
In the following questions, you do not have to perform all numerical calculations, but you have to fully specify the numerical value of each answer.
a. (5 points) Given that the first e-mail you got from sensor S indicates a daily high under 80 degrees, what is the probability that the sensor is placed in Maine?
b. (10 points) Given that the first e-mail you got from sensor S indicates a daily high under 80 degrees, what is the probability that the second e-mail also indicates a daily high under 80 degrees?