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We introduce a general IFS Bayesian method for getting posteriorprobabilities from prior probabilities, and also a generalized Bayes' rule,which will contemplate a dynamical, as well as a non-dynamical setting. Given aloss function ${l}$, we detail the prior and posterior items, theirconsequences and exhibit several examples. Taking $\Theta$ as the set ofparameters and $Y$ as the set of data (which usually provides {randomsamples}), a general IFS is a measurable map $\tau:\Theta\times Y \to Y$, whichcan be interpreted as a family of maps $\tau_\theta:Y\to Y,\,\theta\in\Theta$.The main inspiration for the results we will get here comes from a paper byZellner (with no dynamics), where Bayes' rule is related to a principle ofminimization of {information.} We will show that our IFS Bayesian method whichproduces posterior probabilities (which are associated to holonomicprobabilities) is related to the optimal solution of a variational principle,somehow corresponding to the pressure in Thermodynamic Formalism, and also tothe principle of minimization of information in Information Theory. Among otherresults, we present the prior dynamical elements and we derive thecorresponding posterior elements via the Ruelle operator of ThermodynamicFormalism; getting in this way a form of dynamical Bayes' rule.

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