What does it mean to invert an Exact Renormalization Group Flow?

Building 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, fully information theoretic perspective for understanding ERG through the intermediary of Bayesian Statistical Inference. The 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 continuity equation which we dub Bayesian Diffusion. Identifying the features that define Bayesian Diffusion and mapping them onto the features that define ERG, we obtain a dictionary outlining how ERG can be understood in terms of a statistical inference paradigm run in reverse with an effective coarse-graining imparted by the loss of data. This suggests the compelling interpretation that the inversion of ERG is a form of statistical inference in which the aforementioned lost data is reincorporated into the model. This correspondence matches closely with the Error Correcting Picture of Renormalization, a fact one might have anticipated through the relationship between the Petz Map and the classical Bayesian posterior distribution.