Bayesian Inference for the Multinomial Probit Model under Gaussian Prior Distribution

TL;DR

This paper develops Bayesian inference methods for the multinomial probit model with Gaussian priors, simplifying computations and providing new insights into variational algorithms by leveraging conjugacy properties of skew-normal distributions.

Contribution

It adapts conjugacy results to the multinomial probit model with Gaussian priors, leading to simplified posterior expressions and a novel derivation of variational algorithms.

Findings

Simplified posterior parameter expressions for the multinomial probit model.

New derivation of variational algorithms with computational advantages.

Enhanced understanding of conjugacy in Bayesian multinomial probit models.

Abstract

Multinomial probit (mnp) models are fundamental and widely-applied regression models for categorical data. Fasano and Durante (2022) proved that the class of unified skew-normal distributions is conjugate to several mnp sampling models. This allows to develop Monte Carlo samplers and accurate variational methods to perform Bayesian inference. In this paper, we adapt the abovementioned results for a popular special case: the discrete-choice mnp model under zero mean and independent Gaussian priors. This allows to obtain simplified expressions for the parameters of the posterior distribution and an alternative derivation for the variational algorithm that gives a novel understanding of the fundamental results in Fasano and Durante (2022) as well as computational advantages in our special settings.