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8. Derive the EM algorithm for estimating the Gaussian mixture model $p(X;\theta) = \sum_{k=1}^{K} \pi_k \mathcal{N}(x; \mu_k, \Sigma_k)$, where $\pi_k$ is the mixture weight with $\pi_k \ge 0$ and $\sum_{k=1}^{K} \pi_k = 1$, parameters $\theta = \{\pi_k, \mu_k, \Sigma_k\}_{k=1}^{K}$, and $\mathcal{N}(x; \mu_k, \Sigma_k)$ denotes the value of the Gaussian probability density function with mean $\mu_k$ and covariance matrix $\Sigma_k$ evaluated at $X = x$. In the derivation, first write down the Q function (the E step), and then derive the parameter update rules by maximizing the Q function (the M step). You may resort to any material, but please write down your answer. (L + L)

          8. Derive the EM algorithm for estimating the Gaussian mixture model $p(X;\theta) = \sum_{k=1}^{K} \pi_k \mathcal{N}(x; \mu_k, \Sigma_k)$, where $\pi_k$ is the mixture weight with $\pi_k \ge 0$ and $\sum_{k=1}^{K} \pi_k = 1$, parameters $\theta = \{\pi_k, \mu_k, \Sigma_k\}_{k=1}^{K}$, and $\mathcal{N}(x; \mu_k, \Sigma_k)$ denotes the value of the Gaussian probability density function with mean $\mu_k$ and covariance matrix $\Sigma_k$ evaluated at $X = x$. In the derivation, first write down the Q function (the E step), and then derive the parameter update rules by maximizing the Q function (the M step). You may resort to any material, but please write down your answer. (L + L)

$8. Derive the EM algorithm for estimating the Gaussian mixture model p(X;θ) = ∑k=1^K𝒩(x; , ), where is the mixture weight with ≥ 0 and ∑k=1^K = 1, parameters θ = {, , }k=1^K, and 𝒩(x; , ) denotes the value of the Gaussian probability density function with mean and covariance matrix evaluated at X = x. In the derivation, first write down the Q function (the E step), and then derive the parameter update rules by maximizing the Q function (the M step). You may resort to any material, but please write down your answer. (L + L)$

Added by Christine E.

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