# The d-separation criterion in Categorical Probability

arXiv: https://arxiv.org/abs/2207.05740 · HTML: https://mlsanity.com/paper/2207.05740/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 39 references that are in the map.
- First posted on arXiv 12 Jul 2022.

## Builds on (most cited first, 11 of 11)

- [Causal Networks: Semantics and Expressiveness](https://mlsanity.com/paper/1304.2379/) (2013)
- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Compositional Abstraction Error and a Category of Causal Models](https://mlsanity.com/paper/2103.15758/) (2021)
- [Disintegration and Bayesian Inversion via String Diagrams](https://mlsanity.com/paper/1709.00322/) (2017)
- [Markov Properties for Graphical Models with Cycles and Latent Variables](https://mlsanity.com/paper/1710.08775/) (2017)
- [On the Testability of Causal Models with Latent and Instrumental Variables](https://mlsanity.com/paper/1302.4976/) (2013)
- [Causal Inference by String Diagram Surgery](https://mlsanity.com/paper/1811.08338/) (2018)
- [Instrumentality Tests Revisited](https://mlsanity.com/paper/1301.2258/) (2013)
- [Free gs-monoidal categories and free Markov categories](https://mlsanity.com/paper/2204.02284/) (2022)
- [Rewriting modulo symmetric monoidal structure](https://mlsanity.com/paper/1602.06771/) (2016)
- [Transitional Conditional Independence](https://mlsanity.com/paper/2104.11547/) (2021)

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

- [Causal models in string diagrams](https://mlsanity.com/paper/2304.07638/) (2023)
- [Intuitionistic j-Do-Calculus in Topos Causal Models](https://mlsanity.com/paper/2510.17944/) (2025)
- [A "good regulator theorem" for embodied agents](https://mlsanity.com/paper/2508.06326/) (2025)
- [Active Inference in String Diagrams: A Categorical Account of Predictive Processing and Free Energy](https://mlsanity.com/paper/2308.00861/) (2023)
- [Infinitesimal Causality](https://mlsanity.com/paper/2606.24621/) (2026)
- [A Rose by Any Other Name Would Smell as Sweet: Categorical Homotopy Theory for Large Language Models](https://mlsanity.com/paper/2508.10018/) (2025)
- [Notes on Forr'e's Notion of Conditional Independence and Causal Calculus for Continuous Variables](https://mlsanity.com/paper/2603.24333/) (2026)
- [String Diagrams with Factorized Densities](https://mlsanity.com/paper/2305.02506/) (2023)
- [Topos Causal Models](https://mlsanity.com/paper/2508.08295/) (2025)
- [Consciousness as a Functor](https://mlsanity.com/paper/2508.17561/) (2025)
- [Causal Abstractions, Categorically Unified](https://mlsanity.com/paper/2510.05033/) (2025)
- [Bayesian Networks, Markov Networks, Moralisation, Triangulation: a Categorical Perspective](https://mlsanity.com/paper/2512.09908/) (2025)

## Abstract (arXiv)

The d-separation criterion detects the compatibility of a joint probability distribution with a directed acyclic graph through certain conditional independences. In this work, we study this problem in the context of categorical probability theory by introducing a categorical definition of causal models, a categorical notion of d-separation, and proving an abstract version of the d-separation criterion. This approach has two main benefits. First, categorical d-separation is a very intuitive criterion based on topological connectedness. Second, our results apply both to measure-theoretic probability (with standard Borel spaces) and beyond probability theory, including to deterministic and possibilistic networks. It therefore provides a clean proof of the equivalence of local and global Markov properties with causal compatibility for continuous and mixed random variables as well as deterministic and possibilistic variables.