Emalgorithm

TL;DR

• Iterative method that alternates between estimating missing or latent data (expectation) and updating model parameters to increase data likelihood (maximization).
• Useful for parameter estimation when data are incomplete or contain latent variables.
• Applied to tasks such as estimating means and standard deviations with missing entries, and clustering via estimated cluster memberships and parameters.

Definition

The EMalgorithm is a mathematical technique used in statistics and machine learning to estimate the parameters of a statistical model. It is an iterative method that uses the expectation-maximization (EM) step to update the estimates of the parameters in a way that maximizes the likelihood of the data.

Explanation

The EMalgorithm proceeds by repeating two steps:

• Expectation step: Using the current parameter estimates (for example, means and standard deviations), calculate the expected values of missing or latent data under the assumption that the missing data follow the same distribution as the observed data.
• Maximization step: Use those expected values to update the parameter estimates so as to maximize the likelihood of the observed (and expected) data. This update can be performed with a mathematical optimization technique, such as gradient descent.

A key feature of the EMalgorithm is its ability to make use of all available data even when some observations are missing, enabling more accurate parameter estimates in incomplete-data scenarios.

Examples

Estimating means and standard deviations with missing data

Given a sample containing heights and weights where some entries are missing, the EMalgorithm can estimate the mean and standard deviation of heights and weights by:

• Using current parameter estimates to compute expected values for the missing data (expectation step).
• Updating the mean and standard deviation estimates to maximize the likelihood given those expected values (maximization step).

Clustering data points

For clustering (for example, customer data with age, income, and spending habits), the EMalgorithm can:

• Estimate the probability that each data point belongs to each cluster based on current cluster parameters (means and covariances) in the expectation step.
• Update the cluster parameters to maximize the likelihood of the data, using an optimization technique such as gradient descent, in the maximization step.

Use cases

• Parameter estimation with incomplete or missing data.
• Clustering by estimating cluster memberships and parameters.
• Widely used in statistics, machine learning, and data science.

Notes or pitfalls

• The EMalgorithm relies on the assumption that missing data follow the same distribution as the observed data (as stated in the expectation step).

Related terms

• Expectation step
• Maximization step
• Likelihood function
• Gradient descent
• Clustering
• Mean
• Standard deviation
• Covariance