10. Solve this problem using the EM algorithm: A sample (11, 12, 13) is observed counts...

 

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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? 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?

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a The complete data loglikelihood based on 11 12 21 22 is given by elltheta log binom3811 12 21 22 i... View the full answer