Variational Bayes (VB) methods have emerged as a fast and computationally-efficient alternative to maximum simulated likelihood (MSL) and Markov chain Monte Carlo (MCMC) methods for the estimation of discrete choice models. Thus far, the application of VB has been limited to mixed logit models with unobserved inter-individual taste variation . However, such a model formulation may be too restrictive in panel data settings, as tastes may vary both between individuals as well as across choice tasks  by the same individual. In this paper, we derive a VB method for posterior inference in mixed logit models with unobserved inter- and intra-individual heterogeneity. In a simulation study, we benchmark the performance of the proposed VB method against MSL and MCMC in terms of parameter recovery, predictive accuracy and computational efficiency. The simulation study demonstrates that VB can be a fast and scalable alternative to MSL and MCMC  , especially in applications in which fast predictions are paramount , as VB is observed to be between 2.8 and 17.7 times faster than the two competing methods . Furthermore , the simulation study shows that a parallelised implementation of the MSL estimator with analytical gradients is a viable alternative to MCMC in terms of both estimation accuracy and computational efficiency . In the considered experimental settings, the MSL estimator is  between 0.9 and 2.1 times faster than MCMC.
Variational Bayes (VB) , a method originating from machine learning, enables fast and scalable estimation of complex probabilistic models. Thus far, applications of VB in discrete choice analysis have been limited to mixed logit models with unobserved inter-individual taste heterogeneity . However, such a model formulation may be too restrictive in panel data settings, since tastes may vary both between individuals as well as across choice tasks encountered by the same individual. In this paper, we derive a VB method for posterior inference in mixed logit models with unobserved inter- and intra-individual heterogeneity. In a simulation study, we benchmark the performance of the proposed VB method against maximum simulated likelihood (MSL) and Markov chain Monte Carlo (MCMC) methods in terms of parameter recovery, predictive accuracy and computational efficiency. The simulation study shows that VB can be a fast , scalable and accurate alternative to MSL and MCMC estimation , especially in applications in which fast predictions are paramount . VB is observed to be between 2.8 and 17.7 times faster than the two competing methods , while affording comparable or superior accuracy. Besides , the simulation study demonstrates that a parallelised implementation of the MSL estimator with analytical gradients is a viable alternative to MCMC in terms of both estimation accuracy and computational efficiency , as the MSL estimator is observed to be between 0.9 and 2.1 times faster than MCMC.
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