# Bayesian Renormalization

arXiv: https://arxiv.org/abs/2305.10491 · HTML: https://mlsanity.com/paper/2305.10491/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 103 references that are in the map.
- First posted on arXiv 17 May 2023.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Explaining Neural Scaling Laws](https://mlsanity.com/paper/2102.06701/) (2021)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise](https://mlsanity.com/paper/2208.09392/) (2022)
- [A Review on Generative Adversarial Networks: Algorithms, Theory, and Applications](https://mlsanity.com/paper/2001.06937/) (2020)
- [Why does deep and cheap learning work so well?](https://mlsanity.com/paper/1608.08225/) (2016)
- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [Neural Networks and Quantum Field Theory](https://mlsanity.com/paper/2008.08601/) (2020)
- [An exact mapping between the Variational Renormalization Group and Deep Learning](https://mlsanity.com/paper/1410.3831/) (2014)
- [Building Quantum Field Theories Out of Neurons](https://mlsanity.com/paper/2112.04527/) (2021)
- [Towards quantifying information flows: relative entropy in deep neural networks and the renormalization group](https://mlsanity.com/paper/2107.06898/) (2021)
- [Decomposing neural networks as mappings of correlation functions](https://mlsanity.com/paper/2202.04925/) (2022)
- [The Inverse of Exact Renormalization Group Flows as Statistical Inference](https://mlsanity.com/paper/2212.11379/) (2022)
- [On the Dynamics of Inference and Learning](https://mlsanity.com/paper/2204.12939/) (2022)
- [Coarse--graining, fixed points, and scaling in a large population of neurons](https://mlsanity.com/paper/1809.08461/) (2018)
- [Coarse--graining and hints of scaling in a population of 1000+ neurons](https://mlsanity.com/paper/1812.11904/) (2018)

## Built on by (most cited first, 14 of 14)

- [Renormalizing Diffusion Models](https://mlsanity.com/paper/2308.12355/) (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)
- [Neural Entropy](https://mlsanity.com/paper/2409.03817/) (2024)
- [GUD: Generation with Unified Diffusion](https://mlsanity.com/paper/2410.02667/) (2024)
- [Towards Worst-Case Guarantees with Scale-Aware Interpretability](https://mlsanity.com/paper/2602.05184/) (2026)
- [A path to natural language through tokenisation and transformers](https://mlsanity.com/paper/2601.03368/) (2026)
- [Learning 3-Manifold Triangulations](https://mlsanity.com/paper/2405.09610/) (2024)
- [Lecture Notes on Statistical Physics and Neural Networks](https://mlsanity.com/paper/2605.06394/) (2026)
- [Deep neural networks as lattice gauge theories](https://mlsanity.com/paper/2608.19331/) (2026)
- [Optimal Pruning for Neural Architectures using Fisher Information Distances](https://mlsanity.com/paper/2609.16129/) (2026)

## Abstract (arXiv)

In this note we present a fully information theoretic approach to renormalization inspired by Bayesian statistical inference, which we refer to as Bayesian Renormalization. The main insight of Bayesian Renormalization is that the Fisher metric defines a correlation length that plays the role of an emergent RG scale quantifying the distinguishability between nearby points in the space of probability distributions. This RG scale can be interpreted as a proxy for the maximum number of unique observations that can be made about a given system during a statistical inference experiment. The role of the Bayesian Renormalization scheme is subsequently to prepare an effective model for a given system up to a precision which is bounded by the aforementioned scale. In applications of Bayesian Renormalization to physical systems, the emergent information theoretic scale is naturally identified with the maximum energy that can be probed by current experimental apparatus, and thus Bayesian Renormalization coincides with ordinary renormalization. However, Bayesian Renormalization is sufficiently general to apply even in circumstances in which an immediate physical scale is absent, and thus provides an ideal approach to renormalization in data science contexts. To this end, we provide insight into how the Bayesian Renormalization scheme relates to existing methods for data compression and data generation such as the information bottleneck and the diffusion learning paradigm. We conclude by designing an explicit form of Bayesian Renormalization inspired by Wilson's momentum shell renormalization scheme in Quantum Field Theory. We apply this Bayesian Renormalization scheme to a simple Neural Network and verify the sense in which it organizes the parameters of the model according to a hierarchy of information theoretic importance.