Expectation-Maximization Algorithm

The Expectation-Maximization (EM) algorithm is an iterative optimization algorithm commonly used for estimating parameters in statistical models, especially in the presence of latent variables or missing data. It aims to find the maximum likelihood (ML) or maximum a posteriori (MAP) estimates of parameters when there is incomplete information about the observed data.

The algorithm consists of two main steps:

  1. Expectation (E) Step:
    • In this step, the algorithm computes the expected value (expectation) of the missing or latent variables given the observed data and the current estimate of the model parameters.
    • The E-step involves calculating the posterior probability distribution of the latent variables.
  2. Maximization (M) Step:
    • In this step, the algorithm maximizes the expected log-likelihood obtained from the E-step with respect to the model parameters.
    • The M-step involves finding the parameter values that maximize the expected log-likelihood.

These steps are iteratively repeated until convergence is achieved. The EM algorithm guarantees that with each iteration, the log-likelihood of the observed data increases, or at least does not decrease.

The EM algorithm is widely used in various applications, including:

  • Clustering: In mixture models such as Gaussian Mixture Models (GMMs), the EM algorithm is used to estimate the parameters of each component distribution and the assignment of data points to clusters.
  • Missing Data Imputation: In the presence of missing data, the EM algorithm can be used to impute missing values while simultaneously estimating model parameters.
  • Density Estimation: EM is used for estimating parameters in probabilistic models when the data is incomplete or when there are hidden variables.
  • Biological Sequence Analysis: EM is employed in bioinformatics for tasks such as gene prediction and sequence alignment.

The EM algorithm is a powerful tool for handling complex models with incomplete or latent information, and its applications span various domains within statistics and machine learning.