On the Brittleness of Bayesian Inference

TL;DR

This paper investigates the robustness of Bayesian inference, revealing that it can be brittle in continuous systems with finite information, leading to drastically different conclusions from nearly identical models and data.

Contribution

The study provides new theoretical insights showing Bayesian methods' potential brittleness in continuous settings, highlighting limitations in their robustness under finite information.

Findings

Bayesian inference is robust with finite outcome spaces or limited marginals.

It can be brittle in continuous systems with finite data information.

Small model or prior perturbations can drastically change posterior conclusions.

Abstract

With the advent of high-performance computing, Bayesian methods are increasingly popular tools for the quantification of uncertainty throughout science and industry. Since these methods impact the making of sometimes critical decisions in increasingly complicated contexts, the sensitivity of their posterior conclusions with respect to the underlying models and prior beliefs is a pressing question for which there currently exist positive and negative results. We report new results suggesting that, although Bayesian methods are robust when the number of possible outcomes is finite or when only a finite number of marginals of the data-generating distribution are unknown, they could be generically brittle when applied to continuous systems (and their discretizations) with finite information on the data-generating distribution. If closeness is defined in terms of the total variation metric or the matching of a finite system of generalized moments, then (1) two practitioners who use arbitrarily close models and observe the same (possibly arbitrarily large amount of) data may reach opposite conclusions; and (2) any given prior and model can be slightly perturbed to achieve any desired posterior conclusions. The mechanism causing brittlenss/robustness suggests that learning and robustness are antagonistic requirements and raises the question of a missing stability condition for using Bayesian Inference in a continuous world under finite information.