A Geometric Variational Approach to Bayesian Inference

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∞ in L2 , and the Fisher–Rao metric reduces to the standard L2 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∞ , 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.