  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.
