Fast Variational Bayes Methods for Multinomial Probit Models

Abstract

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

Keywords:

• Large choice datasets
• Multinomial probit model
• Variational inference

Supplementary Materials

The online supplements for “Fast variational Bayes methods for multinomial probit models” area (1) pdf with online appendix and a (2) zip file with the MATLAB code.

This online appendix has six parts:

Part A: Specification of the prior distributions.

Part B: Description of the MCMC sampling algorithm.

Part C: Details of the implementation of VB in the MVMNP model.

Part D: Details of the implementation of VB in the MVMNP model with identity covariance matrix.

Part E: Additional results numerical experiments.

Part F: Additional results empirical applications.

Notes

¹ https://www.dunnhumby.com/source-files/