Fast Variational Bayes Methods for Multinomial Probit Models

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.多项Probit模型（multinomial probit model）常被用于分析选择行为。然而，现有马尔可夫链蒙特卡洛（Markov chain Monte Carlo, MCMC）方法的估计过程计算成本高昂，这限制了其在大规模选择数据集上的应用范围。本文提出了一种精准高效的变分贝叶斯方法，即便在涉及大量选择选项与观测样本的场景下，该方法依然可行。变分方法通常需要为未归一化后验密度构建解析表达式，并合理选择变分族（variational family）。但在多项Probit模型中，这两点均难以设定：其后验分布需要识别约束，且需引入大量潜在效用（latent utilities）进行扩充。我们通过对潜在效用的协方差矩阵实施球面变换，构建出可识别参数的未归一化扩充后验分布，并将潜在效用的条件后验分布作为变分族的组成部分。所提方法的运算速度优于MCMC，且可扩展至大量选择选项与大规模观测样本场景。我们通过数值实验与包含百万级观测样本的真实购买数据集，验证了所提方法的精准性与可扩展性。