Reparameterization invariance in approximate Bayesian inference

Hrittik Roy, Marco Miani, Carl Henrik Ek, Philipp Hennig, Marvin Pförtner, Lukas Tatzel, Søren Hauberg

Keywords: Approximate Bayesian inference, Laplace approximations, Differential geometry, Reparametrization invariance

TL;DR: We provide a mathematical analysis of reparametrizations in the context of Laplace approximations, and devise novel a reparametrization invariant Riemannian diffusion-based approximate posterior.

Abstract: Current approximate posteriors in Bayesian neural networks (BNNs) exhibit a crucial limitation: they fail to maintain invariance under reparameterization, i.e. BNNs assign different posterior densities to different parametrizations of identical functions. This creates a fundamental flaw in the application of Bayesian principles as it breaks the correspondence between uncertainty over the parameters with uncertainty over the parametrized function. In this paper, we investigate this issue in the context of the increasingly popular linearized Laplace approximation. Specifically, it has been observed that linearized predictives alleviate the common underfitting problems of the Laplace approximation. We develop a new geometric view of reparametrizations from which we explain the success of linearization. Moreover, we demonstrate that these reparameterization invariance properties can be extended to the original neural network predictive using a Riemannian diffusion process giving a straightforward algorithm for approximate posterior sampling, which empirically improves posterior fit.

Primary Area: Probabilistic methods (for example: variational inference, Gaussian processes)

Submission Number: 18366