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Published 2015 | Published + Submitted
Journal Article Open

On the Brittleness of Bayesian Inference


With the advent of high-performance computing, Bayesian methods are becoming increasingly popular tools for the quantification of uncertainty throughout science and industry. Since these methods can 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 to which there currently exist positive and negative answers. 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 (TV) 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 conclusion. The mechanism causing brittleness/robustness suggests that learning and robustness are antagonistic requirements, which raises the possibility of a missing stability condition when using Bayesian inference in a continuous world under finite information.

Additional Information

© 2015 Society for Industrial and Applied Mathematics. Received by the editors September 26, 2013; accepted for publication (in revised form) April 9, 2015; published electronically November 5, 2015. The authors gratefully acknowledge support for this work from the Air Force Office of Scientific Research under award FA9550-12-1-0389 (Scientific Computation of Optimal Statistical Estimators). They thank P. Diaconis, D. Mayo, P. Stark, and L. Wasserman for stimulating discussions and relevant references and pointers. They thank the anonymous referees for valuable comments and suggestions.

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Published - 130938633.pdf

Submitted - 1308.6306v3.pdf


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August 20, 2023
October 25, 2023