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

TL;DR

This paper applies large deviations theory to hypothesis testing between singular Gibbs measures, deriving exponential decay rates for error probabilities and extending classical results like Neyman-Pearson, Stein's Lemma, and Chernoff bounds to dynamical systems.

Contribution

It introduces a dynamical version of the Neyman-Pearson lemma for Gibbs measures and explicitly computes decay rates of error probabilities in this context.

Findings

Explicit exponential decay rates for error probabilities.

A dynamical Neyman-Pearson lemma with an optimal test.

Extension of Stein's Lemma and Chernoff bounds to Gibbs measures.

Abstract

We analyze hypotheses tests using classical results on large deviations to compare two models, each one described by a different H\"older Gibbs probability measure. One main difference to the classical hypothesis tests in Decision Theory is that here the two measures are singular with respect to each other. Among other objectives, we are interested in the decay rate of the wrong decisions probability, when the sample size $n$ goes to infinity. We show a dynamical version of the Neyman-Pearson Lemma displaying the ideal test within a certain class of similar tests. This test becomes exponentially better, compared to other alternative tests, when the sample size goes to infinity. We are able to present the explicit exponential decay rate. 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 Chernoff's information are also presented.