The generalized IFS Bayesian method and an associated variational principle covering the classical and dynamical cases
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We introduce a general IFS Bayesian method for getting posterior probabilities from prior probabilities, and also a generalized Bayes' rule, which will contemplate a dynamical, as well as a non dynamical setting. Given a loss function l, we detail the prior and posterior items, their consequences and exhibit several examples. Taking Θ as the set of parameters and Y as the set of data (which usually provides random samples), a general IFS is a measurable map τ:Θ× Y → Y, which can be interpreted as a family of maps τ_θ:Y→ Y, θ∈Θ. The main inspiration for the results we will get here comes from a paper by Zellner (with no dynamics), where Bayes' rule is related to a principle of minimization of information. We will show that our IFS Bayesian method which produces posterior probabilities (which are associated to holonomic probabilities) is related to the optimal solution of a variational principle, somehow corresponding to the pressure in Thermodynamic Formalism, and also to the principle of minimization of information in Information Theory.
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