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Statistical inference can be seen as information processing involving inputinformation and output information that updates belief about some unknownparameters. We consider the Bayesian framework for making inferences aboutdynamical systems from ergodic observations, where the Bayesian procedure isbased on the Gibbs posterior inference, a decision process generalization ofstandard Bayesian inference where the likelihood is replaced by the exponentialof a loss function. In the case of direct observation and almost-additive lossfunctions, we prove an exponential convergence of the a posteriori measures alimit measure. Our estimates on the Bayes posterior convergence for directobservation are related and extend those in a recent paper by K. McGoff, S.Mukherjee and A. Nobel. Our approach makes use of non-additive thermodynamicformalism and large deviation properties instead of joinings.

Bayes posterior convergence for loss functions via almost additive Thermodynamic Formalism