EMAlgorithmusI

EMAlgorithmusI is a form of the Expectation-Maximization (EM) algorithm used for parameter estimation in statistical models with latent variables or incomplete data. It is commonly applied to mixture models, factor analysis, missing-data problems, and other settings where direct maximum likelihood estimation is difficult because some variables are unobserved.

The algorithm iterates two steps until convergence. In the E-step, the expected value of the complete-data log-likelihood is computed with respect to the current parameter estimates, effectively inferring the latent variables by their conditional distribution. In the M-step, this expectation is maximized with respect to the model parameters, producing updated estimates. Formally, Q(θ|θ^t) = E[log L(θ; X, Z) | X, θ^t], where X denotes observed data and Z the latent variables. The E-step uses θ^t to obtain the distribution of Z given X, and the M-step solves θ^{t+1} = argmax_θ Q(θ|θ^t).

As an example, in a Gaussian mixture model with data X and latent component indicators Z, the E-step computes responsibilities γ(z_{nk}) = π_k N(x_n|μ_k, Σ_k) / sum_j π_j N(x_n|μ_j, Σ_j). The M-step then updates π_k, μ_k, and Σ_k using these responsibilities as weights. EMAlgorithmusI guarantees non-decreasing likelihood at each iteration under standard regularity conditions.

Convergence is typically to a local maximum or saddle point and can be sensitive to initialization. Extensions include generalized EM, stochastic EM, and online EM to address speed and robustness. The method was introduced by Dempster, Laird, and Rubin in 1977 and remains a foundational tool for latent-variable models.