Question 1

a) Define the conditional probability p( A | B) in terms of the joint probability p( A, B) . You may want to use a diagram/sketch.

b) Give the law of total probabilities, that is express p( A) using a set of events B1 , B2 ,, BN and the corresponding conditional (or joint) probabilities. What are the conditions that need to hold?

c) Some emails received by users are spams containing viruses. You are building a system to detect such illicit virus emails. You start by using the feature of whether or not an email contains an executable attachment, as this an important datum indicating

d) Explain the difference between Maximum Likelihood (ML) and Maximum a Posterior (MAP) methods of learning parameters theta from data X . whether the email in fact contains a virus. Data analysis suggests that 95% of virus emails contain executable attachments, 90% of legitimate emails do not contain executable attachments, and 2% of emails overall are viruses. i.If your classifier scans an email an executable attachment, what is the probability that the email in fact contains a virus? ii.Comment on this value that you calculate. How does it compare with a decision based only on the frequency of the emails that contain viruses? iii.What is the probability that your classifier makes an error?