The generalized IFS Bayesian method and an associated variational principle covering the classical and dynamical cases

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-dynamicalsetting.Givenalossfunctionl,wedetailthepriorandposterioritems,their consequences and exhibit several examples. Taking Theta as the set of parameters and Y as the set of data (which usually provides random samples), a general IFS is a measurable map tau:circle star xY -> Y, which can be interpreted as a family of maps tau theta:Y -> Y,theta is an element of circle dot. 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 princi-pleofminimizationofinformation.WewillshowthatourIFSBayesianmethod 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. Among other results, we present the prior dynamical elements and we derive the corresponding posterior elements via the Ruelle operator of Thermodynamic Formalism; getting in this way a form of dynamical Bayes' rule.