A Geometric Variational Approach to Bayesian Inference

Abstract

We propose a novel Riemannian geometric framework for variational inference in Bayesian models based on the nonparametric Fisher–Rao metric on the manifold of probability density functions. Under the square-root density representation, the manifold can be identified with the positive orthant of the unit hypersphere $S^{\infty }$ in ${\mathbb{L}}^2$, and the Fisher–Rao metric reduces to the standard ${\mathbb{L}}^2$ metric. Exploiting such a Riemannian structure, we formulate the task of approximating the posterior distribution as a variational problem on the hypersphere based on the α-divergence. This provides a tighter lower bound on the marginal distribution when compared to, and a corresponding upper bound unavailable with, approaches based on the Kullback–Leibler divergence. We propose a novel gradient-based algorithm for the variational problem based on Fréchet derivative operators motivated by the geometry of $S^{\infty }$, and examine its properties. Through simulations and real data applications, we demonstrate the utility of the proposed geometric framework and algorithm on several Bayesian models. Supplementary materials for this article are available online.

Keywords:

• Bayesian density estimation
• Bayesian logistic regression
• Gradient ascent algorithm
• Infinite-dimensional Riemannian optimization
• Square-root density

Acknowledgments

The authors thank Prof. Steven MacEachern for valuable discussions and suggestions. They are also grateful for the comments provided by two anonymous reviewers that improved the contents of this article.

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This research was partially supported by NSF DMS 1613054 and NIH R37 CA214955 (to KB and SK), and NSF CCF 1740761 (to SK).