### Bayesian Renormalization: An explicit correspondence between statistical inference and exact renormalization

Marc Klinger, University of Illinois Urbana-Champaign*Abstract:*

Renormalization is a ubiquitous tool in theoretical physics used to understand the role of scale in organizing natural phenomena. In this presentation we will report on a new information theoretic perspective for understanding the Exact Renormalization Group (ERG) through the intermediary of Bayesian Statistical Inference. This connection is facilitated by the Dynamical Bayesian Inference scheme, which encodes Bayesian inference in the form of a one parameter family of probability distributions solving an integro-differential equation derived from Bayes’ law. Utilizing the picture of an ERG flow as a functional diffusion process, we arrive at a dictionary outlining how renormalization can be understood as an inverse process relative to a Dynamical Bayesian inference scheme. A particularly salient feature of this correspondence is that it identifies the role of Fisher geometry in providing an emergent scale for “Bayesian Renormalization” that is related to the precision with which nearby points in model space can be differentiated. We comment on the usefulness of this identification in data science applications including possible implementations of “Bayesian Renormalization” as a tool for refining diffusion learning techniques.