Gaussian Variational Approximation With a Factor Covariance Structure

Variational approximations have the potential to scale Bayesian computations to large datasets and highly parameterized models. Gaussian approximations are popular, but can be computationally burdensome when an unrestricted covariance matrix is employed and the dimension of the model parameter is high. To circumvent this problem, we consider a factor covariance structure as a parsimonious representation. General stochastic gradient ascent methods are described for efficient implementation, with gradient estimates obtained using the so-called “reparameterization trick.” The end result is a flexible and efficient approach to high-dimensional Gaussian variational approximation. We illustrate using robust P-spline regression and logistic regression models. For the latter, we consider eight real datasets, including datasets with many more covariates than observations, and another with mixed effects. In all cases, our variational method provides fast and accurate estimates. Supplementary material for this article is available online.