Expectation–maximization algorithm

Expectation–maximization (EM) algorithm is a powerful statistical method used for finding maximum likelihood estimates of parameters in probabilistic models, where the model depends on unobserved latent variables. The EM algorithm is an iterative method that alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and a maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the E step. These steps are repeated until convergence.

Overview

The EM algorithm was introduced in a seminal paper by Arthur Dempster, Nan Laird, and Donald Rubin in 1977. It has since become a crucial tool in the field of statistics and machine learning for dealing with incomplete data sets or hidden variables, common in many applications such as speech recognition, computational biology, and image analysis.

Algorithm

The EM algorithm consists of two main steps repeated iteratively:

E-step

During the Expectation step, the algorithm calculates the expected value of the log-likelihood function, with respect to the conditional distribution of the latent variables given the observed data and the current estimates of the parameters. This step involves computing the expectation of the log-likelihood concerning the missing data, treating the parameters as fixed.

M-step

In the Maximization step, the parameters are updated to maximize the expected log-likelihood found in the E-step. This typically involves setting the derivative of the expected log-likelihood with respect to the parameters to zero and solving for the parameters.

Convergence

The EM algorithm is guaranteed to converge to a local maximum (or saddle point) of the likelihood function. Under certain conditions, it can be shown that the sequence of parameter estimates generated by the EM algorithm converges to the set of parameters that maximize the likelihood function.

Applications

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

• Machine Learning: For clustering and density estimation in unsupervised learning.
• Bioinformatics: In the analysis of gene expression data and sequence alignment.
• Image Processing: For image reconstruction and segmentation.
• Speech Recognition: In hidden Markov models for recognizing speech patterns.

Limitations

While the EM algorithm is a powerful tool, it has some limitations:

• It can converge to a local maximum rather than the global maximum of the likelihood function.
• The convergence rate can be slow, especially in the presence of many local maxima.
• Choosing appropriate initial values for the parameters can significantly affect the algorithm's performance.

See also

• Maximum likelihood estimation
• Hidden Markov model
• K-means clustering

References

• Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological).