Scalable Variational Inference for Multinomial Probit Models under Large Choice Sets and Sample Sizes

Abstract: The multinomial probit (MNP) model is widely used to analyze categorical outcomes due to its ability to capture flexible substitution patterns among alternatives. Conventional likelihood based and Markov chain Monte Carlo (MCMC) estimators become computationally prohibitive in high dimensional choice settings. This study introduces a fast and accurate conditional variational inference (CVI) approach to calibrate MNP model parameters, which is scalable to large samples and large choice sets. A flexible variational distribution on correlated latent utilities is defined using neural embeddings, and a reparameterization trick is used to ensure the positive definiteness of the resulting covariance matrix. The resulting CVI estimator is similar to a variational autoencoder, with the variational model being the encoder and the MNP's data generating process being the decoder. Straight through estimation and Gumbel SoftMax approximation are adopted for the argmax operation to select an alternative with the highest latent utility. This eliminates the need to sample from high dimensional truncated Gaussian distributions, significantly reducing computational costs as the number of alternatives grows. The proposed method achieves parameter recovery comparable to MCMC. It can calibrate MNP parameters with 20 alternatives and one million observations in approximately 28 minutes roughly 36 times faster and more accurate than the existing benchmarks in recovering model parameters.