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 $\Theta$ as the set of parameters and $Y$ as the set of data (which usually provides {random samples}), a general IFS is a measurable map $\tau:\Theta\times Y \to Y$, which can 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 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. 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.

翻译：本文提出一种通用的IFS贝叶斯方法，用于从先验概率获取后验概率，并建立广义贝叶斯规则，该规则同时涵盖动态与非动态设定。给定损失函数${l}$，我们详细阐述先验与后验项、其推论，并给出若干实例。以$\Theta$为参数集、$Y$为数据集（通常提供随机样本），一般IFS是一个可测映射$\tau:\Theta\times Y \to Y$，可解释为映射族$\tau_\theta:Y\to Y,\,\theta\in\Theta$。本研究的主要灵感源自Zellner（无动力学情形）的论文，其中贝叶斯规则与信息最小化原理相关联。我们将证明：产生后验概率（与完整概率相关）的IFS贝叶斯方法对应于变分原理的最优解，该变分原理既与热力学形式中的压力对应，又与信息论中的信息最小化原理相关联。在其他成果中，我们给出先验动力元，并通过热力学形式的Ruelle算子推导相应的后验元，从而获得一种动态贝叶斯规则的形式。