Dynamical hypothesis tests and Decision Theory for Gibbs distributions

We consider the problem of testing for two Gibbs probabilities $μ_0$ and $μ_1$ defined for a dynamical system $(Ω,T)$. 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 $h(x_0), h(x_1)=h(T(x_0)),..., h(x_n)=h(T^n(x_0))$, $x_0\in Ω$, $n\ge 0$, where $h:Ω\to\mathbb R$ 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 $Ω$ 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 $n$.

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