# On The Universality of Diagrams for Causal Inference and The Causal Reproducing Property

arXiv: https://arxiv.org/abs/2207.02917 · HTML: https://mlsanity.com/paper/2207.02917/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 73 references that are in the map.
- First posted on arXiv 6 Jul 2022.

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

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Markov Properties for Graphical Models with Cycles and Latent Variables](https://mlsanity.com/paper/1710.08775/) (2017)
- [Margins of discrete Bayesian networks](https://mlsanity.com/paper/1501.02103/) (2015)
- [Bipartite Causal Inference with Interference](https://mlsanity.com/paper/1807.08660/) (2018)
- [A Mathematical Theory of Co-Design](https://mlsanity.com/paper/1512.08055/) (2015)
- [The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits](https://mlsanity.com/paper/1305.0297/) (2013)
- [Online Vector Balancing and Geometric Discrepancy](https://mlsanity.com/paper/1912.03350/) (2019)
- [Universal Decision Models](https://mlsanity.com/paper/2110.15431/) (2021)
- [Causal Homotopy](https://mlsanity.com/paper/2112.01847/) (2021)
- [Generativity and Interactional Effects: an Overview](https://mlsanity.com/paper/1911.10406/) (2019)
- [Causal Inference in Network Economics](https://mlsanity.com/paper/2109.11344/) (2021)

## Built on by (most cited first, 2 of 2)

- [Unifying Causal Inference and Reinforcement Learning using Higher-Order Category Theory](https://mlsanity.com/paper/2209.06262/) (2022)
- [Categoroids: Universal Conditional Independence](https://mlsanity.com/paper/2208.11077/) (2022)

## Abstract (arXiv)

We propose Universal Causality, an overarching framework based on category theory that defines the universal property that underlies causal inference independent of the underlying representational formalism used. More formally, universal causal models are defined as categories consisting of objects and morphisms between them representing causal influences, as well as structures for carrying out interventions (experiments) and evaluating their outcomes (observations). Functors map between categories, and natural transformations map between a pair of functors across the same two categories. Abstract causal diagrams in our framework are built using universal constructions from category theory, including the limit or co-limit of an abstract causal diagram, or more generally, the Kan extension. We present two foundational results in universal causal inference. The first result, called the Universal Causality Theorem (UCT), pertains to the universality of diagrams, which are viewed as functors mapping both objects and relationships from an indexing category of abstract causal diagrams to an actual causal model whose nodes are labeled by random variables, and edges represent functional or probabilistic relationships. UCT states that any causal inference can be represented in a canonical way as the co-limit of an abstract causal diagram of representable objects. UCT follows from a basic result in the theory of sheaves. The second result, the Causal Reproducing Property (CRP), states that any causal influence of a object X on another object Y is representable as a natural transformation between two abstract causal diagrams. CRP follows from the Yoneda Lemma, one of the deepest results in category theory. The CRP property is analogous to the reproducing property in Reproducing Kernel Hilbert Spaces that served as the foundation for kernel methods in machine learning.