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The multinomial probit model is often used to analyze choice behaviour.However, estimation with existing Markov chain Monte Carlo (MCMC) methods iscomputationally costly, which limits its applicability to large choice datasets. This paper proposes a variational Bayes method that is accurate and fast,even when a large number of choice alternatives and observations areconsidered. Variational methods usually require an analytical expression forthe unnormalized posterior density and an adequate choice of variationalfamily. Both are challenging to specify in a multinomial probit, which has aposterior that requires identifying restrictions and is augmented with a largeset of latent utilities. We employ a spherical transformation on the covariancematrix of the latent utilities to construct an unnormalized augmented posteriorthat identifies the parameters, and use the conditional posterior of the latentutilities as part of the variational family. The proposed method is faster thanMCMC, and can be made scalable to both a large number of choice alternativesand a large number of observations. The accuracy and scalability of our methodis illustrated in numerical experiments and real purchase data with one millionobservations.

Fast variational Bayes methods for multinomial probit models