Entropic Inequality Constraints from $e$-separation Relations in Directed Acyclic Graphs with Hidden Variables
This work addresses the challenge of causal inference in systems with unobserved variables, which is a fundamental problem in statistics and machine learning, though it appears incremental as it builds on existing entropic and graphical methods.
The paper tackles the problem of deriving constraints on observed variable distributions in causal models with hidden variables by introducing entropic inequality constraints from e-separation relations in directed acyclic graphs, and it proposes a measure called minimal mediary entropy to augment traditional causal influence metrics.
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.