[Paper Review] The Inverse of Exact Renormalization Group Flows as Statistical Inference

David S. Berman, Marc S. Klinger|arXiv (Cornell University)|Dec 21, 2022

Statistical Mechanics and Entropy69 references4 citations

TL;DR

This paper establishes a duality between Exact Renormalization Group (ERG) flows and Bayesian statistical inference by showing that ERG flows—governed by functional convection-diffusion equations—can be interpreted as the inverse of Dynamical Bayesian Inference, which evolves probability distributions via Bayesian updating. The key contribution is a formal dictionary mapping ERG features to Bayesian inference components, revealing renormalization as information loss and inference as information gain.

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.

Motivation & Objective

• To establish a formal duality between Exact Renormalization Group (ERG) flows and Bayesian statistical inference.
• To reinterpret ERG as a process of information loss through a functional diffusion equation, contrasting it with Bayesian inference as information gain.
• To derive a correspondence between the Dynamical Bayesian Inference equation and the ERG flow via a novel 'Bayesian Diffusion' equation.
• To construct a systematic dictionary mapping ERG components (e.g., beta functions, flow parameters) to Bayesian inference constructs (e.g., likelihoods, priors, posterior updates).
• To demonstrate that renormalizability and scale dependence in ERG correspond to scheme independence and drift in Bayesian inference.

Proposed method

• Formalize the ERG flow as a functional convection-diffusion equation on the space of probability distributions over field configurations.
• Introduce the Dynamical Bayesian Inference framework, which models Bayesian updating as a continuous one-parameter family of probability distributions governed by an integro-differential equation derived from Bayes' law.
• Identify the resulting equation as a 'Bayesian Diffusion' process, a diffusion-type PDE with drift and diffusion terms derived from prior and likelihood.
• Establish equivalence between the Bayesian Diffusion equation and the ERG flow by matching their functional forms and underlying geometric structures.
• Use optimal transport theory to interpret both ERG and Bayesian inference as flows on probability space, with the ERG as forward diffusion and inference as backward reconstruction.
• Derive the Petz map as the mathematical dual of the ERG flow, showing that the inverse of ERG is equivalent to Bayesian inference via the adjoint of a stochastic channel.

Experimental results

Research questions

• RQ1 Can the Exact Renormalization Group (ERG) flow be interpreted as a forward diffusion process in the space of probability distributions?
• RQ2 How does Dynamical Bayesian Inference, which updates beliefs via sequential data, relate to the inverse of ERG flows?
• RQ3 What is the precise mathematical correspondence between the functional convection-diffusion equation of ERG and the Bayesian Diffusion equation?
• RQ4 How do scheme independence in ERG and prior choice in Bayesian inference relate through the structure of the flow equations?
• RQ5 Can renormalizability in quantum field theory be reinterpreted as a stability condition in Bayesian inference, particularly in the late-time limit of the flow?

Key findings

• The ERG flow is mathematically equivalent to a functional convection-diffusion equation, representing a continuous coarse-graining process that loses information.
• Dynamical Bayesian Inference generates a one-parameter family of probability distributions via Bayes' law, which is shown to satisfy a diffusion-type PDE, termed 'Bayesian Diffusion'.
• The Bayesian Diffusion equation is formally equivalent to the ERG flow when the drift and diffusion terms are matched, establishing a duality between inference and renormalization.
• The inverse of an ERG flow corresponds to Bayesian inference, with the initial probability distribution in ERG being reconstructed via Bayesian updating in the inverse process.
• The Petz map, which provides the adjoint of a stochastic channel, is shown to implement the inverse of the ERG flow, confirming the duality at the level of channel theory.
• In the late-time limit, the Bayesian Diffusion equation converges to a stable posterior, analogous to the fixed point of an ERG flow, linking renormalization group fixed points to Bayesian posterior distributions.