Variational Inference for Large-Scale Models of Discrete Choice

Abstract

Discrete choice models are commonly used by applied statisticians in numerous fields, such as marketing, economics, finance, and operations research. When agents in discrete choice models are assumed to have differing preferences, exact inference is often intractable. Markov chain Monte Carlo techniques make approximate inference possible, but the computational cost is prohibitive on the large datasets now becoming routinely available. Variational methods provide a deterministic alternative for approximation of the posterior distribution. We derive variational procedures for empirical Bayes and fully Bayesian inference in the mixed multinomial logit model of discrete choice. The algorithms require only that we solve a sequence of unconstrained optimization problems, which are shown to be convex. One version of the procedures relies on a new approximation to the variational objective function, based on the multivariate delta method. Extensive simulations, along with an analysis of real-world data, demonstrate that variational methods achieve accuracy competitive with Markov chain Monte Carlo at a small fraction of the computational cost. Thus, variational methods permit inference on datasets that otherwise cannot be analyzed without possibly adverse simplifications of the underlying discrete choice model. Appendices C through F are available as online supplemental materials.