Dynamical hypothesis tests and decision theory for Gibbs distributions

Abstract

We consider the problem of testing for two Gibbs probabilities $ \mu_0 $ and $ \mu_1 $ defined for a dynamical system $ (\Omega, T) $. Due to the fact that in general full orbits are not observable or computable, one needs to restrict to subclasses of tests defined by a finite time series $ h(x_0), h(x_1) = h(T(x_0)), ..., h(x_n) = h(T^n(x_0)) $, $ x_0\in \Omega $, $ n\ge 0 $, where $ h:\Omega\to\mathbb R $ denotes a suitable measurable function. We determine in each class the Neyman-Pearson tests, the minimax tests, and the Bayes solutions and show the asymptotic decay of their risk functions as $ n\to\infty $. In the case of $ \Omega $ being a symbolic space, for each $ n\in \mathbb{N} $, these optimal tests rely on the information of the measures for cylinder sets of size $ n $.