Decision Theory and Large Deviations for Dynamical Hypotheses Test: Neyman-Pearson, Min-Max and Bayesian Tests

We analyze hypotheses tests via classical results on large deviations for the case of two different Holder Gibbs probabilities. The main difference for the the classical hypotheses tests in Decision Theory is that here the two considered measures are singular with respect to each other. We analyze the classical Neyman-Pearson test showing its optimality. This test becomes exponentially better when compared to other alternative tests, with the sample size going to infinity. We also consider both, the Min-Max and a certain type of Bayesian hypotheses tests. We shall consider these tests in the log likelihood framework by using several tools of Thermodynamic Formalism. Versions of the Stein's Lemma and the Chernoff's information are also presented.