Assume that X is a random variable with density corresponding to an equal mixture of two...

Assume that X is a random variable with density corresponding to an equal mixture of two Gaussians, with unknown means µ1, µ2 and unknown variances σ1, σ2: p(x) = 0.5N (µ1, σ2 1 ) + 0.5N (µ2, σ2 2 ).

(1) Assume you are given a dataset of n iid samples from this distribution: D = {xi} n i=1. Your goal is to estimate µ1, µ2, σ1 and σ2.

(a) [10 marks] Write down the negative log-likelihood for this problem, for the given dataset D = {xi} n i=1. Simplify as far as you can, by explicitly writing the densities for a Gaussian. Hint: You will not be able to simplify very far, and you will be stuck with a few exponentials that the logs cannot cancel.

(b) [10 marks] Derive update rules to estimate µ1, µ2, σ1, and σ2. More specifically, derive the gradient descent update rule, for the negative log likelihood you provided above.

Expert Solution

orchestra answered 1 year ago

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