Suppose we have a bunch of coins C consisting two kinds of coins. Mathematically, it obeys a mixed Bernoulli distribution: XNF =1rF1+[11r)Fg where 11' E [0,1] and F1 : Ber(p1), F2 2 Ber{p2). That's to say, the kind of the coin is determined by a latent variable Z = 1 or Z = 2. Here Bee-(p) means the coin gives 1 (head) with probability p and gives '1] (tail) with probability 1 p. We initialize parameters p1 = %, pg 2 g and 11' = %. Now, we draw 4 coins X1, X2, X3, X4 independently from C and have 3 independent trials for each of them. The result shows: Coins X1 X2 X3 X4 Trail 1 1 ll D 1 Trail 2 1 1 1 0 Trail 3 ll 1 D 1 (:1) Use EM algorithm for one step, we update F = F(p1 2 %,p2 2 gm = %) to F' Lawn\"). Write down your EM algorithm and show the value of pi, pa, in". Answer: In E step, let 730) = P(X,; ~ Fj), we have: 7(1): WPiU P1) 2 1 1 will p2l + (1 7r)p(1p2) 3 7(3) = Trp1(1 101)2 = 2 1 rpm m) + (1 =r)p2(1 p2l2 3 and a?) =44) = a\" = a, a = as In M step, denote all parameters as 8, let 1:1 : mar; : 1 1r, X ,; is the summation of three trails. Then, we want to optimize (9(6), 851d)