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EM Algorithm for Mixtures of Gaussians [3 P]

Assume that the training examples $ \langle x_i\rangle_{i=1}^N \in \mathbb{R}^d$ were generated from a mixture of $ k$ Gaussian distributions with means $ \mu_j \in \mathbb{R}^d$ and diagonal covariance-matrices $ \Sigma_j =$   diag$ (\sigma_{j1}^2,\ldots, \sigma_{jd}^2) \in \mathbb{R}^{d \times d},  j=1,\ldots,k$ . Extend the EM algorithm for estimating the $ k$ -means and derive an algorithm to estimate also the variances of the Gaussians.

Pfeiffer Michael 2006-01-18