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 L 2 , and the Fisher–Rao metric reduces to the standard 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 ∞ , 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