Students in class take a multiple-choice test with 10 questions and 3 choices per question. A student knowing the answer to a question will answer it correctly, while a student that does not will guess the answer with probability of success 1/3. Each student is equally likely to belong to one of three categories i = 1, 2, 3: those who know the answer to each question with corresponding probabilities theta_i, where theta_1 = 0.3, theta_2 = 0.7, and theta_2= 0.95 (independent of other questions). Suppose that a randomly chosen student answers k questions correctly.
(a) For each possible value of k, derive the MAP estimate of the category that this student belongs to.
(b) Let M be the number of questions that the student knows how to answer. Derive the posterior PMF, and the MAP and LMS estimates of M given that the student answered correctly 5 questions.
I can derive probability density function for each case, but I don't know how it works for "Bayesian Inference", like PMF, MAP and LMS...