The Inverse of Exact Renormalization Group Flows as Statistical Inference

TL;DR

The paper casts Exact Renormalization Group flows as diffusion processes on the space of field-distribution functionals and develops a bridge to Bayesian inference via Dynamical Bayesian Inference and Bayesian Diffusion. It shows that ERG can be interpreted as an inverse process to statistical inference, formalized through a Wasserstein gradient-flow viewpoint and a KL-divergence based energetics, with a concrete dictionary between Wegner-Morris type flows and Bayesian diffusion. By deriving PDEs and FP forms that couple drift and diffusion to log-likelihoods and seed actions, the work provides a principled information-theoretic framework for renormalization, including a notion of renormalizability tied to emergent scales in the Fisher information geometry. The results offer a unifying perspective that connects RG, optimal transport, diffusion learning, holography-inspired ideas, and potential data-science applications, suggesting that inverting RG can be achieved through principled Bayesian reconstruction channels.

Abstract

We build on the view of the Exact Renormalization Group (ERG) as an instantiation of Optimal Transport described by a functional convection-diffusion equation. We provide a new information theoretic perspective for understanding the 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. In this note, we demonstrate how the Dynamical Bayesian Inference equation is, itself, equivalent to a diffusion equation which we dub Bayesian Diffusion. Identifying the features that define Bayesian Diffusion, and mapping them onto the features that define the ERG, we obtain a dictionary outlining how renormalization can be understood as the inverse of statistical inference.