  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.
