Dependency in DAG models with Hidden Variables

TL;DR

This paper characterizes when two variables in DAG models with hidden variables can be identical while remaining independent of others, revealing that such cases occur only when they are densely connected, impacting model search strategies.

Contribution

It provides necessary and sufficient conditions for variable equality in DAG models with hidden variables, highlighting the role of dense connectivity and implications for causal inference.

Findings

Two variables can be identical and independent of others iff they are densely connected.

Densely connected variables can have any joint distribution.

Model search can be simplified by focusing on densely connected vertices.

Abstract

Directed acyclic graph models with hidden variables have been much studied, particularly in view of their computational efficiency and connection with causal methods. In this paper we provide the circumstances under which it is possible for two variables to be identically equal, while all other observed variables stay jointly independent of them and mutually of each other. We find that this is possible if and only if the two variables are `densely connected'; in other words, if applications of identifiable causal interventions on the graph cannot (non-trivially) separate them. As a consequence of this, we can also allow such pairs of random variables have any bivariate joint distribution that we choose. This has implications for model search, since it suggests that we can reduce to only consider graphs in which densely connected vertices are always joined by an edge.