Common Algorithm for Parameter Estimation - EM (Expectation-Maximization)

The Expectation-Maximization (EM) Algorithm is a classical probabilistic model parameter estimation method, particularly suitable for scenarios involving latent variables or missing data. Its core concept addresses incomplete data problems—where traditional maximum likelihood estimation struggles—through iterative optimization.

The algorithm primarily consists of two alternating steps: the E-step (Expectation step) and the M-step (Maximization step). During the E-step, the algorithm computes the conditional probability expectation of latent variables based on current parameter estimates. In code implementation, this typically involves calculating posterior probabilities using functions like `scipy.stats` for probability distributions. During the M-step, these expected values are used to re-estimate model parameters, maximizing the likelihood function to achieve local optima. This iterative process continuously refines parameter estimates until convergence to a stable solution.

The EM algorithm is widely applied in probabilistic graphical models with latent variables, such as Gaussian Mixture Models (GMMs) and Hidden Markov Models (HMMs). Common implementations leverage libraries like `scikit-learn` (e.g., `GaussianMixture` class) or custom log-likelihood functions. Compared to direct optimization of complex likelihood functions, EM decomposes the problem into two more manageable sub-steps, significantly improving computational feasibility. Notably, the algorithm is sensitive to initial values and generally guarantees convergence only to local optima, which can be mitigated using multiple random initializations in practice.