# Resolution of Simpson's paradox via the common cause principle

arXiv: https://arxiv.org/abs/2403.00957 · HTML: https://mlsanity.com/paper/2403.00957/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 60 references that are in the map.
- First posted on arXiv 1 Mar 2024.

## Builds on (most cited first, 5 of 5)

- [Simpson's paradox in Covid-19 case fatality rates: a mediation analysis of age-related causal effects](https://mlsanity.com/paper/2005.07180/) (2020)
- [Can you Trust the Trend: Discovering Simpson's Paradoxes in Social Data](https://mlsanity.com/paper/1801.04385/) (2018)
- [Computational Social Scientist Beware: Simpson's Paradox in Behavioral Data](https://mlsanity.com/paper/1710.08615/) (2017)
- [The most likely common cause](https://mlsanity.com/paper/2306.17557/) (2023)
- [The role of exchangeability in causal inference](https://mlsanity.com/paper/2006.01799/) (2020)

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

- [Nonnegative matrix factorization and the principle of the common cause](https://mlsanity.com/paper/2509.03652/) (2025)

## Abstract (arXiv)

Simpson's paradox is an obstacle to establishing a probabilistic association between two events a1 and a2, given the third (lurking) random variable B. We focus on scenarios when the random variables A (which combines a1, a2, and their complements) and B have a common cause C that need not be observed. Alternatively, we can assume that C screens out A from B. For such cases, the correct association between a1 and a2 is to be defined via conditioning over C. This setup generalizes the original Simpson's paradox: now its two contradicting options refer to two particular and different causes C. We show that if B and C are binary and A is quaternary (the minimal and the most widespread situation for the Simpson's paradox), the conditioning over any binary common cause C establishes the same direction of association between a1 and a2 as the conditioning over B in the original formulation of the paradox. Thus, for the minimal common cause, one should choose the option of Simpson's paradox that assumes conditioning over B and not its marginalization. The same conclusion is reached when Simpson's paradox is formulated via 3 continuous Gaussian variables: within the minimal formulation of the paradox (3 scalar continuous variables A1, A2, and B), one should choose the option with the conditioning over B.