Problem 1: EM Algorithm for the Gaussian Mixture Model (25 pts) In the following Gaussian Mixture Model Xi|zi = 0 ~ N(μ0, σ0^2); Xi|zi = 1 ~ N(μ1, σ1^2); zi ~ Bernoulli(π), i = 1, ..., n, where Xi's are observable random variables, and zi's are hidden random variables. Given observed data points x1, ..., xn; derive the EM algorithm for estimating μ0, μ1, σ0^2, σ1^2 and π in the following steps:
In the M-step of the (t + 1)-th iteration; derive the updated parameter estimates μ0^(t+1), μ1^(t+1), (σ0^2)^(t+1), (σ1^2)^(t+1), π^(t+1) based on γi1, ..., γin and τi1, ..., τin.