Expectation Maximization

Expectation maximization is an iterative algorithm used for finding maximum likelihood estimates of parameters in statistical models, particularly when the data is incomplete or has missing values.

Expectation maximization is commonly used in machine learning and statistics for clustering, density estimation, and handling missing data. It is particularly useful when the data is clustered around two or more local centers.

Expectation maximization works by iteratively performing two steps: the Expectation step (E-step) and the Maximization step (M-step). In the E-step, the algorithm calculates the expected value of the latent variables given the observed data and current parameter estimates. In the M-step, the algorithm maximizes the likelihood function to update the parameter estimates based on the expected values calculated in the E-step. These steps are repeated until convergence.

For example, consider a dataset of customer purchase behaviors where some data points are missing. Expectation maximization can be used to estimate the missing values and cluster customers into different segments based on their purchasing patterns.

Expectation maximization is similar to the K-means algorithm in that both require a predefined number of clusters and use iterative optimization to find cluster centers. However, expectation maximization differs in that it assumes a multivariate Gaussian distribution of data points around cluster centers and provides probabilities for each data point belonging to each cluster, whereas K-means assigns each data point to a single cluster.

Common applications of expectation maximization include Gaussian Mixture Models (GMMs), where it is used to estimate the parameters of the mixture components, and handling missing data in datasets.