What is em?

EM, short for Expectation-Maximization, is an iterative algorithm used to find maximum likelihood estimates of parameters in statistical models, where the model depends on unobserved latent variables. It's particularly useful in scenarios involving incomplete data or mixture models.

Here's a breakdown of key aspects:

• Purpose: To estimate parameters when some data is missing or hidden, or when the model involves a mixture of distributions. This is crucial in situations where direct optimization of the likelihood function is difficult or impossible.

• The Two Steps: The EM algorithm consists of two alternating steps:

  • Expectation (E) Step: This step computes the expected value of the log-likelihood function, using the current estimate of the parameters. It essentially "fills in" the missing data using the best available estimates.
  • Maximization (M) Step: This step finds the parameter values that maximize the expected log-likelihood computed in the E-step. This step updates the parameter estimates based on the "completed" data.

• Iteration: The E and M steps are repeated iteratively until the parameter estimates converge, or a stopping criterion is met (e.g., a small change in the likelihood or parameters).

• Applications: EM is widely used in various fields, including:

  • Clustering (e.g., Gaussian Mixture Models)
  • Image processing
  • Natural language processing
  • Bioinformatics (e.g., sequence alignment)
  • Hidden Markov Models

• Advantages:

  • Guaranteed to converge to a local maximum of the likelihood function.
  • Relatively easy to implement.
  • Handles missing data effectively.

• Disadvantages:

  • Can be sensitive to initial parameter values.
  • Convergence can be slow.
  • Only guarantees convergence to a local maximum, not necessarily the global maximum. Therefore, multiple runs with different initializations are often recommended.
  • Requires knowledge of the underlying statistical model.

The EM algorithm provides a powerful tool for parameter estimation in the presence of incomplete data, making it a valuable technique in various statistical modeling applications.