Expectation Maximization and Posterior Constraints

The expectation maximization (EM) algorithm is a widely used maximum likelihood estimationprocedure for statistical models when the values of some of the variables in the model are not observed. Very often, however, our aim is primarily tofind a model that assigns values to the latent variables that have intended meaning for our data and maximizing expected likelihood only sometimes accomplishes this.Unfortunately, it is typically difficult to add even simple a-priori information about latent variables in graphical models without making the models overly complex or intractable. In this paper, we present an efficient, principled way to inject rich constraints on the posteriors of latent variables into the EM algorithm. Our method can be used to learn tractable graphical models that satisfy additional,otherwise intractable constraints. Focusing on clustering and the alignment problem for statistical machine translation, we show that simple, intuitive posteriorconstraints can greatly improve the performance over standard baselines and be competitive with more complex, intractable models.