A univariate Gaussian Mixture Model (GMM) has two components, both of which have their own mean and standard deviation. The model is defined by the following parameters:
Z Bernoulli(theta) p(x|z = 0) ~ N(u, sigma) p(x|z = 1) ~ N (2u, 3sigma)
For a dataset of N data points, find the following:
2.1.1. Write the marginal probability of x, i.e. p(x) [2pts]
2.1.2. E-Step: Compute the posterior probability, i.e., p(zi = k|x'), where k = {0,1} [2pts]
2.1.3. M-Step: Compute the updated value of u (You can keep o fixed for this) [3pts]
2.1.4. M-Step: Compute the updated value for sigma (You can keep u fixed for this) [3pts]