# Factored space models: Towards causality between levels of abstraction

arXiv: https://arxiv.org/abs/2412.02579 · HTML: https://mlsanity.com/paper/2412.02579/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 29 references that are in the map.
- First posted on arXiv 3 Dec 2024.

## Builds on (most cited first, 13 of 13)

- [Language Models (Mostly) Know What They Know](https://mlsanity.com/paper/2207.05221/) (2022)
- [Causal Abstractions of Neural Networks](https://mlsanity.com/paper/2106.02997/) (2021)
- [Causal Consistency of Structural Equation Models](https://mlsanity.com/paper/1707.00819/) (2017)
- [Abstracting Causal Models](https://mlsanity.com/paper/1812.03789/) (2018)
- [Metacognitive Capabilities of LLMs: An Exploration in Mathematical Problem Solving](https://mlsanity.com/paper/2405.12205/) (2024)
- [Robust agents learn causal world models](https://mlsanity.com/paper/2402.10877/) (2024)
- [Approximate Causal Abstraction](https://mlsanity.com/paper/1906.11583/) (2019)
- [Agent Incentives: A Causal Perspective](https://mlsanity.com/paper/2102.01685/) (2021)
- [Discovering Agents](https://mlsanity.com/paper/2208.08345/) (2022)
- [Abstraction between Structural Causal Models: A Review of Definitions and Properties](https://mlsanity.com/paper/2207.08603/) (2022)
- [Causal inference via algebraic geometry: feasibility tests for functional causal structures with two binary observed variables](https://mlsanity.com/paper/1506.03880/) (2015)
- [Causal Models with Constraints](https://mlsanity.com/paper/2301.06845/) (2023)
- [Temporal Inference with Finite Factored Sets](https://mlsanity.com/paper/2109.11513/) (2021)

## Built on by (most cited first, 1 of 1)

- [Causal Foundations of Collective Agency](https://mlsanity.com/paper/2605.00248/) (2026)

## Abstract (arXiv)

Causality plays an important role in understanding intelligent behavior, and there is a wealth of literature on mathematical models for causality, most of which is focused on causal graphs. Causal graphs are a powerful tool for a wide range of applications, in particular when the relevant variables are known and at the same level of abstraction. However, the given variables can also be unstructured data, like pixels of an image. Meanwhile, the causal variables, such as the positions of objects in the image, can be arbitrary deterministic functions of the given variables. Moreover, the causal variables may form a hierarchy of abstractions, in which the macro-level variables are deterministic functions of the micro-level variables. Causal graphs are limited when it comes to modeling this kind of situation. In the presence of deterministic relationships there is generally no causal graph that satisfies both the Markov condition and the faithfulness condition. We introduce factored space models as an alternative to causal graphs which naturally represent both probabilistic and deterministic relationships at all levels of abstraction. Moreover, we introduce structural independence and establish that it is equivalent to statistical independence in every distribution that factorizes over the factored space. This theorem generalizes the classical soundness and completeness theorem for d-separation.