Discrete choice models describe the choices made by decision makers among alternatives and play an important role in transportation planning, marketing research and other applications. The mixed multinomial logit (MMNL) model is a popular discrete choice model that captures heterogeneity in the preferences of decision makers through random coefficients. While Markov chain Monte Carlo methods provide the Bayesian analogue to classical procedures for estimating MMNL models, computations can be prohibitively expensive for large datasets. Approximate inference can be obtained using variational methods at a lower computational cost with competitive accuracy. In this paper, we develop variational methods for estimating MMNL models that allow random coefficients to be correlated in the posterior and can be extended easily to large-scale datasets. We explore three alternatives: (1) Laplace variational inference, (2) nonconjugate variational message passing and (3) stochastic linear regression. Their performances are compared using real and simulated data. To accelerate convergence for large datasets, we develop stochastic variational inference for MMNL models using each of the above alternatives. Stochastic variational inference allows data to be processed in minibatches by optimizing global variational parameters using stochastic gradient approximation. A novel strategy for increasing minibatch sizes adaptively within stochastic variational inference is proposed.
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