Question

3. Consider a Gaussian mixture model with K component Gaussians with different means having the same covariance matrix Σ but all How would you modify the equations of the Expectation Maximization algorithm in order to take into account the fact that the covariance matrix is the same for all components? Justify your answer.

Answer 1

A Gaussian mixture model is parameterized by two types of values, the mixture component weights and the component means and variances/covariances. For a Gaussian mixture model with K components, the
th
component has a mean of $\mu _k$ and variance of $\sigma _k$ for the uni-variate case and a mean of $\mu_k ^\rightarrow$ and covariance matrix of $\sum_k$ for the multivariate case. The mixture component weights are defined as $\phi _k$ for component $C_k$, with the constraint that $\sum_{i=1}^{K} \phi _i = 1$ . so that the total probability distribution normalizes to 1. If the component weights aren't learned, they can be viewed as an a-priori distribution over components such that p(x generated by component
Ck)
= $\phi_k.$. If they are instead learned, they are the a-posteriori estimates of the component probabilities given the data.

one-dinensonal Moeld: 꼭,F Muli-alimensinal mdel CS