Owed to a Martingale: A Fast Bayesian On-Line EM Algorithm for Multinomial Models

This paper introduces a fast Bayesian online expectation maximization (BOEM) algorithm for multinomial mixtures. Using some properties of the Dirichlet distribution, we derive expressions for adaptive learning rates that depend solely on the data and the prior’s hyperparameters. As a result, we avoid the problem of having to tune the learning rates using heuristics. In the application to multinomial clustering, choosing the prior’s hyperparameters is an easy task. Our experiments on large real data sets demonstrate that our Bayesian online learning algorithms are fast and provide accurate regularized solutions. We prove asymptotic convergence of our algorithms using stochastic approximation theory.