# Compositional Abstraction Error and a Category of Causal Models

arXiv: https://arxiv.org/abs/2103.15758 · HTML: https://mlsanity.com/paper/2103.15758/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 47 references that are in the map.
- First posted on arXiv 29 Mar 2021.

## Builds on (most cited first, 12 of 12)

- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Causal Consistency of Structural Equation Models](https://mlsanity.com/paper/1707.00819/) (2017)
- [Abstracting Causal Models](https://mlsanity.com/paper/1812.03789/) (2018)
- [Approximate Causal Abstraction](https://mlsanity.com/paper/1906.11583/) (2019)
- [Visual Causal Feature Learning](https://mlsanity.com/paper/1412.2309/) (2014)
- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Causal Inference by String Diagram Surgery](https://mlsanity.com/paper/1811.08338/) (2018)
- [When the map is better than the territory](https://mlsanity.com/paper/1612.09592/) (2016)
- [Causal Discovery from Subsampled Time Series Data by Constraint Optimization](https://mlsanity.com/paper/1602.07970/) (2016)
- [Causal Feature Learning for Utility-Maximizing Agents](https://mlsanity.com/paper/2005.08792/) (2020)
- [MERLiN: Mixture Effect Recovery in Linear Networks](https://mlsanity.com/paper/1512.01255/) (2015)
- [The algebra and machine representation of statistical models](https://mlsanity.com/paper/2006.08945/) (2020)

## Built on by (most cited first, 30 of 33)

- [Finding Alignments Between Interpretable Causal Variables and Distributed Neural Representations](https://mlsanity.com/paper/2303.02536/) (2023)
- [Causal Abstraction: A Theoretical Foundation for Mechanistic Interpretability](https://mlsanity.com/paper/2301.04709/) (2023)
- [Causal Abstraction with Soft Interventions](https://mlsanity.com/paper/2211.12270/) (2022)
- [Jointly Learning Consistent Causal Abstractions Over Multiple Interventional Distributions](https://mlsanity.com/paper/2301.05893/) (2023)
- [Abstraction between Structural Causal Models: A Review of Definitions and Properties](https://mlsanity.com/paper/2207.08603/) (2022)
- [Targeted Reduction of Causal Models](https://mlsanity.com/paper/2311.18639/) (2023)
- [Causal models in string diagrams](https://mlsanity.com/paper/2304.07638/) (2023)
- [Causal Optimal Transport of Abstractions](https://mlsanity.com/paper/2312.08107/) (2023)
- [The d-separation criterion in Categorical Probability](https://mlsanity.com/paper/2207.05740/) (2022)
- [Context-Specific Causal Discovery for Categorical Data Using Staged Trees](https://mlsanity.com/paper/2106.04416/) (2021)
- [Learning Structural Causal Models through Deep Generative Models: Methods, Guarantees, and Challenges](https://mlsanity.com/paper/2405.05025/) (2024)
- [Quantifying Consistency and Information Loss for Causal Abstraction Learning](https://mlsanity.com/paper/2305.04357/) (2023)
- [Interventionally Consistent Surrogates for Agent-based Simulators](https://mlsanity.com/paper/2312.11158/) (2023)
- [Causally Abstracted Multi-armed Bandits](https://mlsanity.com/paper/2404.17493/) (2024)
- [Aligning Graphical and Functional Causal Abstractions](https://mlsanity.com/paper/2412.17080/) (2024)
- [What is causal about causal models and representations?](https://mlsanity.com/paper/2501.19335/) (2025)
- [Meaningful Causal Aggregation and Paradoxical Confounding](https://mlsanity.com/paper/2304.11625/) (2023)
- [Towards Computing an Optimal Abstraction for Structural Causal Models](https://mlsanity.com/paper/2208.00894/) (2022)
- [Combining Causal Models for More Accurate Abstractions of Neural Networks](https://mlsanity.com/paper/2503.11429/) (2025)
- [Distributionally Robust Causal Abstractions](https://mlsanity.com/paper/2510.04842/) (2025)
- [Modeling Discrimination with Causal Abstraction](https://mlsanity.com/paper/2501.08429/) (2025)
- [Identifiability in Causal Abstractions: A Hierarchy of Criteria](https://mlsanity.com/paper/2507.06213/) (2025)
- [Axiomatization of Interventional Probability Distributions](https://mlsanity.com/paper/2305.04479/) (2023)
- [CASR: Refining Action Segmentation via Marginalizing Frame-levle Causal Relationships](https://mlsanity.com/paper/2311.12401/) (2023)
- [On the Identifiability of Causal Abstractions](https://mlsanity.com/paper/2503.10834/) (2025)
- [Learning Nonlinear Causal Reductions to Explain Reinforcement Learning Policies](https://mlsanity.com/paper/2507.14901/) (2025)
- [Causal Feature Learning in the Social Sciences](https://mlsanity.com/paper/2503.12784/) (2025)
- [Multi-Level Causal Embeddings](https://mlsanity.com/paper/2602.22287/) (2026)
- [Causal Foundations of Collective Agency](https://mlsanity.com/paper/2605.00248/) (2026)
- [Validating Causal Abstraction Metrics on Simulated Complex Systems](https://mlsanity.com/paper/2607.00267/) (2026)

## Abstract (arXiv)

Interventional causal models describe several joint distributions over some variables used to describe a system, one for each intervention setting. They provide a formal recipe for how to move between the different joint distributions and make predictions about the variables upon intervening on the system. Yet, it is difficult to formalise how we may change the underlying variables used to describe the system, say moving from fine-grained to coarse-grained variables. Here, we argue that compositionality is a desideratum for such model transformations and the associated errors: When abstracting a reference model M iteratively, first obtaining M' and then further simplifying that to obtain M'', we expect the composite transformation from M to M'' to exist and its error to be bounded by the errors incurred by each individual transformation step. Category theory, the study of mathematical objects via compositional transformations between them, offers a natural language to develop our framework for model transformations and abstractions. We introduce a category of finite interventional causal models and, leveraging theory of enriched categories, prove the desired compositionality properties for our framework.