Problem 1: Bayes Suppose that we take the Dirichlet distribution over the probabilities p ~ Dir(01, 02, ..., On) for of > 0. Recall that the Dirichlet probability density function takes the form f(P1 , . . . . Pn) = B(Q) i= 1 You just need to know that B(o) = Spo . Son o lip." dpi . . . dpa. Now suppose that you observe the random vector Y E N" which is a bag of words where the j" coordinate specifies the number of times that word j appears in a document. We can define Y as Y = Wi where w; is the random vector such that exactly one coordinate is 1 while all others are 0 and (w;)() = 1 if word i is j. We take all w; to be independent and we assume that the probability that P((wi)() = 1) = p;, again recalling that at any time word i can only be one specific word. Actual problem: Compute the posterior distribution of the p given Y = y. That is compute the probability density f (p1 = v1, . . . ; Pn = Un |Y =y) Recall that (PI = VI; . . . ; Pn = Un Y = y) = ( =y P1 = v1, . . . ;Pn = Un)f (P1 = v1, . . . , Pn = Un) P(Y = y) However, note that P(Y = y) does not depend on the numerical values v1, ..., Un. Therefore, P(p1 = v1, . . . , Pn = Un|Y = y) x P(Y = ylp1 = v1, . . . ; Pn = Un)f(P1 = v1, . . . , Pn = Un) So to compute the true distribution you just need to calculate the normalizing constant Z = J P(Y = y|p1 = v1, . . .; Pn = Un)f(P1 = v1, .. .; Pn = Un)du . . . dun which of course is equal to P(Y = y). However, by identifying an appropriate pattern, you shouldn't need to compute the integral (nor can you)