  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.
