  In this paper we consider 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. This work establishes tight connections
between Gibbs posterior inference and the thermodynamic formalism, which may
inspire new proof techniques in the study of Bayesian posterior consistency for
dependent processes.
