多项式概率模型的快速变分Bayes方法
Fast variational Bayes methods for multinomial probit models
摘要：
多项概率模型经常被用来分析选择行为。然而，估计与现有的马尔可夫链蒙特卡罗（MCMC）方法是计算昂贵的，这限制了其适用性大的选择数据集。本文提出了一种变分贝叶斯方法，是准确和快速的，即使在大量的选择方案和观察被认为是。变分方法通常要求非归一化后验密度的解析表达式和适当的变分族选择。这两个都是具有挑战性的指定在一个多项概率，这有一个后验，需要确定的限制，并增加了一个大的潜在效用。我们采用球面变换的协方差矩阵的潜在效用，以构建一个非归一化的增广后验，确定的参数，并使用条件后验的潜在效用的变分家庭的一部分。该方法比MCMC更快，并且可以扩展到大量的选择方案和大量的观察。我们的方法的准确性和可扩展性的数值实验和实际购买数据与一百万观察说明。
英文摘要：
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.
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