Dynamical hypothesis tests and Decision Theory for Gibbs distributions

TL;DR

This paper develops hypothesis testing methods for Gibbs distributions in dynamical systems, focusing on finite data and providing optimal tests with asymptotic risk decay, especially in symbolic spaces.

Contribution

It introduces a framework for Neyman-Pearson, minimax, and Bayes tests for Gibbs measures based on finite time series data, with asymptotic analysis.

Findings

Optimal tests depend on cylinder set measures in symbolic spaces.

Asymptotic decay of risk functions as data length increases.

Explicit characterization of Neyman-Pearson, minimax, and Bayes solutions.

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 $n$.