[9_1_B]

Please answer this question step by step

1. Consider the problem of estimating the probability distribution of a scalar random variable x given a set of samples: X1, X2, ..., Xn. (a) Let the desired probability distribution be p(x; a), where a is a parameter of the functional form of p. Describe the procedure of using the maximum-likelihood method to estimate a. (b) Let the desired probability distribution be a 1-D normal distribution with mean u and standard deviation o. Derive the maximum-likelihood estimation of u. Note: Use log-likelihood. (c) A more complex probability distribution can be estimated as a mixture of parameterized distributions. Write down the 1-D probability distribution as a mixture of Gaussians. (d) Explain the concept of "hidden" or "latent" variables in this problem. (e) The standard algorithm to estimate the parameters in a mixture model is the EM algorithm. What do the terms E and M stand for? You have to describe their meanings. Show the steps of the EM algorithm with pseudo-code; you don't need to give the actual equations