Reparameterization invariance in approximate Bayesian inference

TL;DR

This paper addresses the lack of reparameterization invariance in approximate Bayesian neural network posteriors, proposing a geometric approach and a Riemannian diffusion method to improve posterior sampling and fit.

Contribution

It introduces a geometric perspective on reparameterization invariance and develops a Riemannian diffusion process for better approximate Bayesian inference in neural networks.

Findings

Linearized Laplace approximation benefits from reparameterization invariance.

The proposed Riemannian diffusion method empirically improves posterior fit.

The geometric view explains the success of linearization in BNNs.

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.