# Probabilistic Reasoning across the Causal Hierarchy

arXiv: https://arxiv.org/abs/2001.02889 · HTML: https://mlsanity.com/paper/2001.02889/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jan 2020.

## Built on by (most cited first, 19 of 19)

- [Neural Causal Models for Counterfactual Identification and Estimation](https://mlsanity.com/paper/2210.00035/) (2022)
- [Counterfactual Identifiability of Bijective Causal Models](https://mlsanity.com/paper/2302.02228/) (2023)
- [Comparing Causal Frameworks: Potential Outcomes, Structural Models, Graphs, and Abstractions](https://mlsanity.com/paper/2306.14351/) (2023)
- [A Topological Perspective on Causal Inference](https://mlsanity.com/paper/2107.08558/) (2021)
- [A Measure-Theoretic Axiomatisation of Causality](https://mlsanity.com/paper/2305.17139/) (2023)
- [Learning Structural Causal Models through Deep Generative Models: Methods, Guarantees, and Challenges](https://mlsanity.com/paper/2405.05025/) (2024)
- [What is causal about causal models and representations?](https://mlsanity.com/paper/2501.19335/) (2025)
- [Position: Causal Machine Learning Requires Rigorous Synthetic Experiments for Broader Adoption](https://mlsanity.com/paper/2508.08883/) (2025)
- [The Hardness of Reasoning about Probabilities and Causality](https://mlsanity.com/paper/2305.09508/) (2023)
- [Formalizing Statistical Causality via Modal Logic](https://mlsanity.com/paper/2210.16751/) (2022)
- [On Probabilistic and Causal Reasoning with Summation Operators](https://mlsanity.com/paper/2405.03069/) (2024)
- [Nondeterministic Causal Models](https://mlsanity.com/paper/2405.14001/) (2024)
- [Probabilistic Modelling is Sufficient for Causal Inference](https://mlsanity.com/paper/2512.23408/) (2025)
- [Observing Interventions: A logic for thinking about experiments](https://mlsanity.com/paper/2111.12978/) (2021)
- [Probabilistic and Causal Satisfiability: Constraining the Model](https://mlsanity.com/paper/2504.19944/) (2025)
- [Teleporter Theory: A General and Simple Approach for Modeling Cross-World Counterfactual Causality](https://mlsanity.com/paper/2406.11501/) (2024)
- [Learning Counterfactual Outcomes Under Rank Preservation](https://mlsanity.com/paper/2502.06398/) (2025)
- [Gateways to Tractability for Satisfiability in Pearl's Causal Hierarchy](https://mlsanity.com/paper/2511.08091/) (2025)
- [Certifying Interventional Agreement Among Observationally Equivalent Causal Models](https://mlsanity.com/paper/2609.32247/) (2026)

## Abstract (arXiv)

We propose a formalization of the three-tier causal hierarchy of association, intervention, and counterfactuals as a series of probabilistic logical languages. Our languages are of strictly increasing expressivity, the first capable of expressing quantitative probabilistic reasoning -- including conditional independence and Bayesian inference -- the second encoding do-calculus reasoning for causal effects, and the third capturing a fully expressive do-calculus for arbitrary counterfactual queries. We give a corresponding series of finitary axiomatizations complete over both structural causal models and probabilistic programs, and show that satisfiability and validity for each language are decidable in polynomial space.