variationalEM

Variational EM is an algorithmic framework that extends the Expectation-Maximization (EM) method by using variational inference to handle intractable latent-variable posteriors. In standard EM, the E-step computes the exact posterior distribution of latent variables given the data and current parameters. In Variational EM, this step is replaced by optimizing a variational distribution q(Z) to approximate the posterior p(Z|X,θ).

The key idea is to maximize a lower bound on the observed-data log-likelihood, known as the Evidence Lower Bound (ELBO). The ELBO is L(q, θ) = E_q[log p(X,Z|θ) − log q(Z)]. By optimizing q, we tighten the bound on the log-likelihood; by optimizing θ, we increase the bound for the current q. In practice, the algorithm alternates between updating q (the variational E-step) and updating θ (the M-step) so that the ELBO does not decrease.

Variational EM is particularly useful when the exact E-step is intractable or computationally expensive, but a tractable variational family for q(Z) can be chosen (for example, mean-field approximations). It allows approximate inference in complex models such as mixtures with complicated latent structure, topic models, hierarchical models, and other latent-variable formulations where exact posterior computations are prohibitive.

Relation to other methods: Variational EM is often described as a bridge between EM and variational Bayes, and it is sometimes referred to as VB-EM. Unlike fully Bayesian treatments that place priors over parameters and integrate them out, Variational EM focuses on obtaining a tractable, optimistic optimization of the likelihood with an approximate posterior over latent variables.

Convergence properties resemble EM in that each iteration increases or leaves unchanged the ELBO, though results may depend on the chosen variational family and initialization.