# A Geometric Variational Approach to Bayesian Inference

arXiv: https://arxiv.org/abs/1707.09714 · HTML: https://mlsanity.com/paper/1707.09714/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jul 2017.

## Built on by (most cited first, 2 of 2)

- [Geometric variational inference](https://mlsanity.com/paper/2105.10470/) (2021)
- [Variational Inference Optimized Using the Curved Geometry of Coupled Free Energy](https://mlsanity.com/paper/2506.09091/) (2025)

## Abstract (arXiv)

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 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 alpha-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 Frechet derivative operators motivated by the geometry of the Hilbert sphere, 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.