Title: Partially factorized and tighter variational Bayes for probit models
Authors: Daniele Durante - Bocconi University (Italy) [presenting]
Augusto Fasano - Bocconi University (Italy)
Abstract: Bayesian regression models for dichotomous data arise in several applications. Within such a framework, recent research has shown that the posterior distribution for the $p$ probit coefficients has a unified skew-normal kernel, under Gaussian priors, and hence can be expressed via a convolution of $p$-variate Gaussians and $n$-variate truncated normals with full covariance matrix. Such a novel result allows efficient Bayesian inference for a wide class of applications, but closed-form calculation of posterior moments and predictive distributions is unfeasible for large sample sizes due to the intractability of multivariate truncated normals with dependent components. To address this issue we propose a variational approximation for the unified skew-normal posterior which factorizes the multivariate truncated Gaussian component via a product of univariate truncated normals. We prove that such a result can be formally interpreted as a partially factorized mean-field variational Bayes strategy which provides a tighter approximation to the posterior distribution for the probit coefficients, compared to state-of-the-art solutions, while crucially preserving skewness. A simple coordinate ascent variational inference algorithm is developed and the improved performance is outlined in simulations and applications.