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Density estimation 1 #card #exam_question #density_estimation
+ Given the following set of data points in a two-dimensional case, we want to build a probability distribution that represents them. For this aim, we have chosen a mixture model with two Gaussians, $\sum\limits_{k=1}^2(\pi_k \mathcal N (x|\mu_k , \sigma_k^2 I)$, and to optimize its parameters we will k=1 use the EM algorithm. The initial random means for the Gaussians are given by $\mu_1$ and $\mu_2$ and the covariance matrices are initially equal, but constrained to the form $σ_k^2 I$.
+ a) Explain the EM algorithm steps and draw on the figure approximately the component parameters (the mean and the shape of the covariance) after one interation. ( Points)
+ b) Derive the maximum likelihood re-estimation of the component parameters $\sigma_k$ given a set of data points. $\mathcal N (x|\mu, \Sigma) = \frac{1}{(2\pi)D/2 |\Sigma|^{1/2}} e^{−1/2(x − \mu)^T \Sigma^{−1} (x − \mu)}$
+ c) Explain the difference between the re-estimation of the mean of the clusters in the K-means algorithm and the EM algorithm.


- Density estimation 1:
#exam_question #density_estimation
Given the following set of data points in a two-dimensional case, we want to build a probability distribution that represents them. For this aim, we have chosen a mixture model with two Gaussians, $\sum\limits_{k=1}^2(\pi_k \mathcal N (x|\mu_k , \sigma_k^2 I)$, and to optimize its parameters we will k=1 use the EM algorithm. The initial random means for the Gaussians are given by $\mu_1$ and $\mu_2$ and the covariance matrices are initially equal, but constrained to the form $σ_k^2 I$.

a) Explain the EM algorithm steps and draw on the figure approximately the component parameters (the mean and the shape of the covariance) after one interation. ( Points)

b) Derive the maximum likelihood re-estimation of the component parameters $\sigma_k$ given a set of data points.
$\mathcal N (x|\mu, \Sigma) = \frac{1}{(2\pi)D/2 |\Sigma|^{1/2}} e^{−1/2(x − \mu)^T \Sigma^{−1} (x − µ)}$

c) Explain the difference between the re-estimation of the mean of the clusters in the K-means algorithm and the EM algorithm.


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