# The Inverse of Exact Renormalization Group Flows as Statistical Inference

arXiv: https://arxiv.org/abs/2212.11379 · HTML: https://mlsanity.com/paper/2212.11379/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 81 references that are in the map.
- First posted on arXiv 21 Dec 2022.

## Builds on (most cited first, 7 of 7)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Deep Bayesian Inversion](https://mlsanity.com/paper/1811.05910/) (2018)
- [On the Dynamics of Inference and Learning](https://mlsanity.com/paper/2204.12939/) (2022)
- [Probabilistic Numerical Methods for Partial Differential Equations and Bayesian Inverse Problems](https://mlsanity.com/paper/1605.07811/) (2016)

## Built on by (most cited first, 12 of 12)

- [FAdam: Adam is a natural gradient optimizer using diagonal empirical Fisher information](https://mlsanity.com/paper/2405.12807/) (2024)
- [Bayesian Renormalization](https://mlsanity.com/paper/2305.10491/) (2023)
- [Rigor with Machine Learning from Field Theory to the Poincar'e Conjecture](https://mlsanity.com/paper/2402.13321/) (2024)
- [On learning higher-order cumulants in diffusion models](https://mlsanity.com/paper/2410.21212/) (2024)
- [Bayesian RG Flow in Neural Network Field Theories](https://mlsanity.com/paper/2405.17538/) (2024)
- [NCoder -- A Quantum Field Theory approach to encoding data](https://mlsanity.com/paper/2402.00944/) (2024)
- [Wilsonian Renormalization of Neural Network Gaussian Processes](https://mlsanity.com/paper/2405.06008/) (2024)
- [Fermions and Supersymmetry in Neural Network Field Theories](https://mlsanity.com/paper/2511.16741/) (2025)
- [Grokking vs. Learning: Same Features, Different Encodings](https://mlsanity.com/paper/2502.01739/) (2025)
- [A path to natural language through tokenisation and transformers](https://mlsanity.com/paper/2601.03368/) (2026)
- [Lecture Notes on Statistical Physics and Neural Networks](https://mlsanity.com/paper/2605.06394/) (2026)
- [Optimal Pruning for Neural Architectures using Fisher Information Distances](https://mlsanity.com/paper/2609.16129/) (2026)

## Abstract (arXiv)

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.