Variational inference for large-scale models of discrete choice

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 data sets 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. Extensive simulations demonstrate that variational methods achieve accuracy competitive with Markov chain Monte Carlo, at a small fraction of the computational cost. Thus, variational methods permit inferences on data sets that otherwise could not be analyzed without bias-inducing modifications to the underlying model.