Fast variational Bayes methods for multinomial probit models

TL;DR

This paper introduces a fast variational Bayes approach for multinomial probit models, enabling efficient analysis of large choice datasets by overcoming computational limitations of traditional MCMC methods.

Contribution

It develops a novel variational Bayes method using spherical transformation and latent utility conditioning to improve speed and scalability in multinomial probit model estimation.

Findings

Method is faster than MCMC for large datasets

Achieves accurate estimation with high-dimensional choice data

Demonstrated on real purchase data with one million observations

Abstract

The multinomial probit model is often used to analyze choice behaviour. However, estimation with existing Markov chain Monte Carlo (MCMC) methods is computationally costly, which limits its applicability to large choice data sets. This paper 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.