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

Abstract

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 $\tau :\mathrm{\Theta }\times Y\to Y$, which can be interpreted as a family of maps ${\tau }_{\theta }:Y\to Y,\theta \in \mathrm{\Theta }$. 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.

Keywords:

• Generalized Baye's rule
• posterior probability
• general IFS Bayesian method
• minimization of information
• holonomic probability
• Thermodynamic Formalism

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 If there are more than one stationary ρ with respect to $(\overline{l},\nu ,\tau )$, then we get more than one possible posterior probability π

2 observe the importance of considering $d\tilde{\pi }={\tilde{\pi }}_p(\theta )\mathrm{d\theta d}{\delta }_{y_0}$, to get the below expression from (32(32) $\begin{array}{r}\underset{\tilde{\pi }\mathrm{holonomic}}{sup}\int [\log (l(\theta ,y))+\log ({\pi }_a(\theta ))-\log (\phi (y))]\mathrm{d}\tilde{\pi }+H^{\mathrm{d\theta }}(\tilde{\pi }).\end{array}$(32) ), and that the supremum is over ${\tilde{\pi }}_p(\theta )$, which is a probability density function and no more a probability; furthermore, for the last term we take into account (31(31) $\begin{array}{r}{H}^{\nu }(\pi )=\{\begin{array}{ll}-\int \log (J)\mathrm{d}\pi & \mathrm{if}\mathrm{d}\pi =J(\theta ,y)\mathrm{d}\nu (\theta )\mathrm{d}\rho (y)\\ -\mathrm{\infty }& \begin{array}{l}\mathrm{if}\pi \text{is not absolutely continuous}\\ \text{with respect to}\nu \times \rho \end{array}\end{array}\end{array}$(31) ).