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What problem does the Expectation-Maximization algorith...
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See plansEM is for maximum-likelihood (or MAP) parameter estimation when your model has latent variables you can’t observe — cluster assignments in a Gaussian mixture, missing feature values, or a hidden node in a Bayesian network with no labels. Directly maximizing the likelihood is hard because it requires marginalizing over the unobserved variable, which usually makes the log-likelihood non-convex and analytically intractable.
EM sidesteps this by alternating two steps until convergence. In the E-step, given the current parameter estimates, compute the expected value (posterior distribution) of the latent variables — for a Gaussian mixture this means computing the soft cluster-membership probability of every point under the current means and covariances. In the M-step, treat those expectations as if they were true values and re-estimate the parameters that maximize the now-tractable expected complete-data log-likelihood — for the mixture, a weighted mean and covariance update using the soft assignments as weights. Each EM iteration is guaranteed to never decrease the observed-data log-likelihood, so it converges monotonically, but only to a local optimum — the result depends on initialization, which is why in practice you run EM from multiple random starts (or a smarter init like k-means++) and keep the best result. k-means itself is a hard-assignment special case of EM under a Gaussian mixture with fixed spherical covariance.