Dynamics of Bayesian Updating with Dependent Data and Misspecified Models

By Cosma Rohilla Shalizi. 2009

156 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a415b37eb4d0936b3b3f5957232e223c30784233

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Citations in the corpus

Power-Law Distributions in Empirical Data (2007, methodology): “The Bayesian estimation used is equivalent to a smoothing, which to some extent buffers the results against the effects of fluctuations [52], but the method itself is not capable of determining whether the results could be due to chance [38, 64].”
Thermodynamics as a theory of decision-making with information-processing costs (2012, background): “Furthermore, both bounded rational controllers and Bayes’ rule share their structural form with discrete replicator dynamics that model evolutionary processes [85], where samples (a population) are pushed through a fitness function (likelihood, gain function) that biases the distribution of the popu”
Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It (2014, background): “settings (Shalizi, 2009), and more specific settings (Sriram et al.”
Unrealistic Expectations and Misguided Learning (2018, methodology): “To prove convergence, we cannot apply results from the statistical learning literature, such as those of Berk (1966) and Shalizi (2009), where the observer does not choose actions.”
Selectionist and Evolutionary Approaches to Brain Function: A Critical Appraisal (2012, background): “A few people (Zhang, 1999; Harper, 2009; Shalizi, 2009) have realized the connection in formal terms.”
Posterior Consistency of Bayesian Quantile Regression Based on the Misspecified Asymmetric Laplace Density (2013, unknown): “Shalizi (2009) considers general non i.i.d. case.”
Inference Compilation and Universal Probabilistic Programming (2016, background): “A limitation that comes with not learning the generative model itself—as is done by the models organized around the variational autoencoder (Kingma and Welling, 2014; Burda et al., 2016)—is the possibility of model misspeciﬁcation (Shalizi et al., 2009; Gel-man and Shalizi, 2013).”