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Activity Number: 7 - Fiber Bundles in Statistical Inference and Probability
Type: Invited
Date/Time: Sunday, July 28, 2019 : 2:00 PM to 3:50 PM
Sponsor: IMS
Abstract #300451 Presentation
Title: Gibbs Posterior Consistency and the Thermodynamic Formalism
Author(s): Kevin McGoff* and Andrew B Nobel and Sayan Mukherjee
Companies: UNC Charlotte and University of North Carolina at Chapel Hill and Duke University

In this talk, I will discuss recent work (joint with Sayan Mukherjee and Andrew Nobel) on a Bayesian framework for making inferences about dynamical systems from ergodic observations. The proposed Bayesian procedure is based on the Gibbs posterior, a decision theoretic generalization of standard Bayesian inference. We place a prior over a model class consisting of a parametrized family of Gibbs measures on a mixing shift of finite type. This model class generalizes (hidden) Markov chain models by allowing for long range dependencies, including Markov chains of arbitrarily large orders. We characterize the asymptotic behavior of the Gibbs posterior distribution on the parameter space as the number of observations tends to infinity. In particular, we define a limiting variational problem over the space of joinings of the model system with the observed system, and we show that the Gibbs posterior distributions concentrate around the solution set of this variational problem. In the case of properly specified models our convergence results may be used to establish posterior consistency. The talk will highlight the important role played by fiber entropy in the variational problem.

Authors who are presenting talks have a * after their name.

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