Dynamical hypothesis tests and Decision Theory for Gibbs distributions
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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∈Ω, n≥ 0, where h:Ω→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→∞. In the case of Ω being a symbolic space, for each n∈ℕ, these optimal tests rely on the information of the measures for cylinder sets of size n.
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