= their model. For our purposes, it can be described as a multinomial model with the four categories C1,C2, C3, and C4.

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Their Model For Our Purposes It Can Be Described As A Multinomial Model With The Four Categories C1 C2 C3 And C4 1 (75.11 KiB) Viewed 30 times

= their model. For our purposes, it can be described as a multinomial model with the four categories C1,C2, C3, and C4. For a sample of size n, let X = (X1, X2, X3, X4)' denote the observed frequencies of the four categories. Hence, n = {i=1 Xį. The probability model is = i=1 C1 C2 C3 C4 3 + 10 | 1-20 | 1-20 | 10 where the parameter 0 satisfies 0 <0 < 1. In this exercise, we obtain the mle of 0.

6.6.2. In this exercise, we set up an EM algorithm to determine the mle for the situation described in Exercise 6.6.1. Split category Cį into the two subcategories C11 and C12 with probabilities 1/2 and 0/4, respectively. Let Z11 and Z12 denote the respective "frequencies.” Then X1 211 + Z12. Of course, we cannot observe 211 and 212. Let Z = (Z11, Z12)'. = (a) Obtain the complete likelihood Lº(@|x, z). (b) Using the last result and (6.6.22), show that the conditional pmf k(z|0, x) is binomial with parameters Xı and probability of success 0/(2+0). (c) Obtain the E step of the EM algorithm given an initial estimate ô(0) of 0. That is, obtain Q[0@0), x) = Egolog L*(0|X, Z) 60), x). (l ] = Recall that this expectation is taken using the conditional pmf k(zlão), x). Keep in mind the next step; i.e., we need only terms that involve 0. (d) For the M step of the EM algorithm, solve the equation Q(0]@0), x)/30 = 0. Show that the solution is B1), x1@0) + 2x4 + xx@co) nő0) + 2(x2 + x3 + x4) (6.6.23)