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2009 Dynamics of Bayesian updating with dependent data and misspecified models
Cosma Rohilla Shalizi
Electron. J. Statist. 3(none): 1039-1074 (2009). DOI: 10.1214/09-EJS485

Abstract

Much is now known about the consistency of Bayesian updating on infinite-dimensional parameter spaces with independent or Markovian data. Necessary conditions for consistency include the prior putting enough weight on the correct neighborhoods of the data-generating distribution; various sufficient conditions further restrict the prior in ways analogous to capacity control in frequentist nonparametrics. The asymptotics of Bayesian updating with mis-specified models or priors, or non-Markovian data, are far less well explored. Here I establish sufficient conditions for posterior convergence when all hypotheses are wrong, and the data have complex dependencies. The main dynamical assumption is the asymptotic equipartition (Shannon-McMillan-Breiman) property of information theory. This, along with Egorov’s Theorem on uniform convergence, lets me build a sieve-like structure for the prior. The main statistical assumption, also a form of capacity control, concerns the compatibility of the prior and the data-generating process, controlling the fluctuations in the log-likelihood when averaged over the sieve-like sets. In addition to posterior convergence, I derive a kind of large deviations principle for the posterior measure, extending in some cases to rates of convergence, and discuss the advantages of predicting using a combination of models known to be wrong. An appendix sketches connections between these results and the replicator dynamics of evolutionary theory.

Citation

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Cosma Rohilla Shalizi. "Dynamics of Bayesian updating with dependent data and misspecified models." Electron. J. Statist. 3 1039 - 1074, 2009. https://doi.org/10.1214/09-EJS485

Information

Published: 2009
First available in Project Euclid: 29 October 2009

zbMATH: 1326.62017
MathSciNet: MR2557128
Digital Object Identifier: 10.1214/09-EJS485

Subjects:
Primary: 62C10, 62G20, 62M09
Secondary: 60F10, 62M05, 92D15, 94A17

Rights: Copyright © 2009 The Institute of Mathematical Statistics and the Bernoulli Society

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