Notes on Semi-Supervised Expectation Maximization

This work considers the Expectation Maximization (EM) algorithm in the semi-supervised setting. First, the general form for semi-supervised version of maximum likelihood is derived from the Latent Variable Model (LVM). Since the involved integrals are usually intractable, a surrogate objective function based on the Evidence Lower Bound (ELBO) is introduced. Next, we derive the equations of the semi-supervised EM. Finally, the concrete equations for a fitting a Gaussian Mixture Model (GMM) using labeled and unlabeled data are deduced.