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