Bayesian Estimation of Mixed Multinomial Logit Models: Advances and Simulation-Based Evaluations

Variational Bayes (VB) methods have emerged as a fast and computationally-efficient alternative to Markov chain Monte Carlo (MCMC) methods for Bayesian estimation of mixed multinomial logit (MMNL) models. It has been established that VB is substantially faster than MCMC at practically no compromises in predictive accuracy. In this paper, we address two critical gaps concerning the usage and understanding of VB for MMNL. First, extant VB methods are limited to utility specifications involving only individual-specific taste parameters. Second, the finite-sample properties of VB estimators and the relative performance of VB, MCMC and maximum simulated likelihood estimation (MSLE) are not known. To address the former, this study extends several VB methods for MMNL to admit utility specifications including both fixed and random utility parameters. To address the latter, we conduct an extensive simulation-based evaluation to benchmark the extended VB methods against MCMC and MSLE in terms of estimation times, parameter recovery and predictive accuracy. The results suggest that all VB variants perform as well as MCMC and MSLE at prediction and recovery of all model parameters with the exception of the covariance matrix of the multivariate normal mixing distribution. In particular, VB with nonconjugate variational message passing and the delta-method (VB-NCVMP-Delta) is relatively accurate and up to 15 times faster than MCMC and MSLE. On the whole, VB-NCVMP-Delta is most suitable for applications in which fast predictions are paramount, while MCMC should be preferred in applications in which accurate inferences are most important.