# Disintegration and Bayesian Inversion via String Diagrams

arXiv: https://arxiv.org/abs/1709.00322 · HTML: https://mlsanity.com/paper/1709.00322/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Aug 2017.

## Built on by (most cited first, 30 of 32)

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- [A Category-theoretical Meta-analysis of Definitions of Disentanglement](https://mlsanity.com/paper/2305.06886/) (2023)
- [Active Inference in String Diagrams: A Categorical Account of Predictive Processing and Free Energy](https://mlsanity.com/paper/2308.00861/) (2023)
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- [Compositional Semantics for Probabilistic Programs with Exact Conditioning](https://mlsanity.com/paper/2101.11351/) (2021)
- [Compositional Active Inference II: Polynomial Dynamics. Approximate Inference Doctrines](https://mlsanity.com/paper/2208.12173/) (2022)
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- [A Markov Categorical Framework for Language Modeling](https://mlsanity.com/paper/2507.19247/) (2025)
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- [A Rose by Any Other Name Would Smell as Sweet: Categorical Homotopy Theory for Large Language Models](https://mlsanity.com/paper/2508.10018/) (2025)
- [A Probabilistic Generative Model of Free Categories](https://mlsanity.com/paper/2205.04545/) (2022)
- [Formalising and Learning a Quantum Model of Concepts](https://mlsanity.com/paper/2302.14822/) (2023)
- [String Diagrams with Factorized Densities](https://mlsanity.com/paper/2305.02506/) (2023)
- [Pearl's and Jeffrey's Update as Modes of Learning in Probabilistic Programming](https://mlsanity.com/paper/2309.07053/) (2023)
- [Overdrawing Urns using Categories of Signed Probabilities](https://mlsanity.com/paper/2312.12453/) (2023)
- [Topos Causal Models](https://mlsanity.com/paper/2508.08295/) (2025)
- [Consciousness as a Functor](https://mlsanity.com/paper/2508.17561/) (2025)

## Abstract (arXiv)

The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability --- via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.