# Entropic Inequality Constraints from e-separation Relations in Directed Acyclic Graphs with Hidden Variables

arXiv: https://arxiv.org/abs/2107.07087 · HTML: https://mlsanity.com/paper/2107.07087/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 39 references that are in the map.
- First posted on arXiv 15 Jul 2021.

## Builds on (most cited first, 10 of 10)

- [Differentiable Causal Discovery Under Unmeasured Confounding](https://mlsanity.com/paper/2010.06978/) (2020)
- [Nested Markov Properties for Acyclic Directed Mixed Graphs](https://mlsanity.com/paper/1701.06686/) (2017)
- [Ordering-Based Causal Structure Learning in the Presence of Latent Variables](https://mlsanity.com/paper/1910.09014/) (2019)
- [Fast Causal Inference with Non-Random Missingness by Test-Wise Deletion](https://mlsanity.com/paper/1705.09031/) (2017)
- [A Potential Outcomes Calculus for Identifying Conditional Path-Specific Effects](https://mlsanity.com/paper/1903.03662/) (2019)
- [The Inflation Technique for Causal Inference with Latent Variables](https://mlsanity.com/paper/1609.00672/) (2016)
- [Multivariate Counterfactual Systems And Causal Graphical Models](https://mlsanity.com/paper/2008.06017/) (2020)
- [Quantifier Elimination for Statistical Problems](https://mlsanity.com/paper/1301.6698/) (2013)
- [Smooth, identifiable supermodels of discrete DAG models with latent variables](https://mlsanity.com/paper/1511.06813/) (2015)
- [Inferring latent structures via information inequalities](https://mlsanity.com/paper/1407.2256/) (2014)

## Built on by (most cited first, 3 of 3)

- [On Testability of the Front-Door Model via Verma Constraints](https://mlsanity.com/paper/2203.00161/) (2022)
- [Classifying Causal Structures: Ascertaining when Classical Correlations are Constrained by Inequalities](https://mlsanity.com/paper/2308.02380/) (2023)
- [On the physics of nested Markov models: a generalized probabilistic theory perspective](https://mlsanity.com/paper/2411.11614/) (2024)

## Abstract (arXiv)

Directed acyclic graphs (DAGs) with hidden variables are often used to characterize causal relations between variables in a system. When some variables are unobserved, DAGs imply a notoriously complicated set of constraints on the distribution of observed variables. In this work, we present entropic inequality constraints that are implied by e-separation relations in hidden variable DAGs with discrete observed variables. The constraints can intuitively be understood to follow from the fact that the capacity of variables along a causal pathway to convey information is restricted by their entropy; e.g. at the extreme case, a variable with entropy 0 can convey no information. We show how these constraints can be used to learn about the true causal model from an observed data distribution. In addition, we propose a measure of causal influence called the minimal mediary entropy, and demonstrate that it can augment traditional measures such as the average causal effect.