related to MachineLearning Problem

(1) Assume that X is a random variable with density corresponding to an equal mixture of two Gaussians, with unknown means H1, H2 and unknown variances 01, 02: p(x) = 0.5N (111,09) +0.5N (42, ). Assume you are given a dataset of n iid samples from this distribution: D = {}=1. Your goal is to estimate 41, 42,01 and 02. (a) [10 MARKS) Write down the negative log-likelihood for this problem, for the given dataset D= {}-1. Simplify as far as you can, by explicitly writing the densities for a Gaussian. (b) [10 MARKS] Derive update rules to estimate M1, M2,01, and 02. More specifically, derive the gradient descent update rule, for the negative log likelihood you provided above. (1) Assume that X is a random variable with density corresponding to an equal mixture of two Gaussians, with unknown means H1, H2 and unknown variances 01, 02: p(x) = 0.5N (111,09) +0.5N (42, ). Assume you are given a dataset of n iid samples from this distribution: D = {}=1. Your goal is to estimate 41, 42,01 and 02. (a) [10 MARKS) Write down the negative log-likelihood for this problem, for the given dataset D= {}-1. Simplify as far as you can, by explicitly writing the densities for a Gaussian. (b) [10 MARKS] Derive update rules to estimate M1, M2,01, and 02. More specifically, derive the gradient descent update rule, for the negative log likelihood you provided above