A comparison of variational approximations for fast inference in mixed logit models

Variational Bayesian methods aim to address some of the weaknesses (computation time, storage costs and convergence monitoring) of mainstream MCMC-based inference at the cost of a biased approximation to the posterior distribution. We investigate the performance of variational approximations in the context of the mixed logit model, which is arguably one of the most used models for discrete choice data. A typical treatment using the variational Bayesian methodology is hindered by the fact that the expectation of the so called log-sum-exponential function has no closed form expression. Therefore, one has to resort to approximating or bounding this term. In this paper we compare seven di_erent possible bounds or approximations. We found that quadratic bounds do not perform particularly well. A recently proposed non-quadratic bound, on the other hand, did perform quite well. We also found that the approximation used in a previous study only performed well for speci_c settings. Our proposed approximation based on quasi Monte Carlo sampling on the other hand performed consistently well across all simulation settings while remaining computationally tractable.