= = 1. (A) Consider a simple random sample X1, ...,Xn drawn from a K-class Gaussian mixture model with the following lat

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1 A Consider A Simple Random Sample X1 Xn Drawn From A K Class Gaussian Mixture Model With The Following Lat 1 (100.5 KiB) Viewed 28 times

= = 1. (A) Consider a simple random sample X1, ...,Xn drawn from a K-class Gaussian mixture model with the following latent variable representation: Pr(Z = k) = Tk, X|Z=k~ N(uk, o), k= 1,...,K, where {Teks Mlik, o2}K1 are unknown. In order to complete this exercise effectively, please carefully review our “Mixture Model” lecture notes; note that for your responses, you'll need to provide any proof details omitted by the lecture notes. (a) Find the marginal density of X. Find the mean and variance of X. (b) Based on X1, ... ,Xn, find the log-likelihood of {Tk, Mk, 0%}K=1: (c) Let Z; denote the latent variable for Xị. Compute Pr(Zi = k | Xị). (d) Assume that Pr(Zi = k | Xị) is a known constant for all i = 1,...,n and all k 1,...,K. Find optimal values of {Tek, Mx, o3}K-1 that maximize the log-likelihood from Part (b). = =