10. Solve this problem using the EM algorithm: A sample (11, 12, 13) is observed counts from a multinomial population with probabilities: (-10, 10, 10+1). The objective is to obtain the maximum likelihood estimate of 0. The pdf of multinomial distribution for this sample is p(1; 0) In order to use EM algorithm, we put this into the framework of an incomplete data problem. Define (11, 12, 21, 22) with multinomial distribution probabilities (-10,10,10,1), where 2 + 22 = 13, (21, 22) are missing data. Consider the estimation of when (11,12,13) (38, 34, 125), do the following: 71 '1 12 n! es!!es! ( 2 30 ) * (10) ** (10 + )* . 1 = (a) Write down the complete data log-likelihood based on (11, 12, 21, 22). (b) Describe the steps in the EM algorithm to compute the EM estimate of 0. For example, what are the E-step and M-step equations, and how to iterate the algorithm until it converges. [Hint: e.g., conditioning on z+z2 = 13, 21 is distributed as Binomial (125, p = (0/4)/(1/2+0/4)). Note: Even if you have a closed-form solution, still do it using this algorithm]. (c) Write an R. program to realize the algorithm in Part (b). Include your R code and report the result you obtain, i.e., what is your initial value in iteration, what is the estimate of 0, and to what decimal place that your algorithm stops?