Dynamics of Bayesian updating with dependent data and misspecified models
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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.