Brittleness of Bayesian Inference Under Finite Information in a Continuous World

TL;DR

This paper investigates the limitations of Bayesian inference in continuous settings with finite data, revealing inherent brittleness and the antagonistic relationship between learning and robustness through a novel measure optimization calculus.

Contribution

It introduces a reduction calculus for measure optimization problems, providing bounds on Bayesian posterior predictions and analyzing the stability and brittleness of Bayesian models under finite information.

Findings

Bayesian models can exhibit maximal prediction error despite large data samples.

Learning and robustness are fundamentally antagonistic properties.

A new measure calculus helps understand the mechanisms behind brittleness.

Abstract

We derive, in the classical framework of Bayesian sensitivity analysis, optimal lower and upper bounds on posterior values obtained from Bayesian models that exactly capture an arbitrarily large number of finite-dimensional marginals of the data-generating distribution and/or that are as close as desired to the data-generating distribution in the Prokhorov or total variation metrics; these bounds show that such models may still make the largest possible prediction error after conditioning on an arbitrarily large number of sample data measured at finite precision. These results are obtained through the development of a reduction calculus for optimization problems over measures on spaces of measures. We use this calculus to investigate the mechanisms that generate brittleness/robustness and, in particular, we observe that learning and robustness are antagonistic properties. It is now well understood that the numerical resolution of PDEs requires the satisfaction of specific stability conditions. Is there a missing stability condition for using Bayesian inference in a continuous world under finite information?