The most likely common cause

TL;DR

This paper explores identifying hidden common causes in observed variables using generalized maximum likelihood, revealing phase transition-like behavior and complex causal structures beyond traditional DAG models.

Contribution

It introduces a generalized maximum likelihood approach for latent confounder identification and discusses its relation to entropy and phase transition phenomena.

Findings

Generalized likelihood can identify common causes consistent with the principle.

Discovered phase transition behavior in conditional probabilities for binary variables.

Uncovered complex causal structures beyond DAG representations.

Abstract

The common cause principle for two random variables $A$ and $B$ is examined in the case of causal insufficiency, when their common cause $C$ is known to exist, but only the joint probability of $A$ and $B$ is observed. As a result, $C$ cannot be uniquely identified (the latent confounder problem). We show that the generalized maximum likelihood method can be applied to this situation and allows identification of $C$ that is consistent with the common cause principle. It closely relates to the maximum entropy principle. Investigation of the two binary symmetric variables reveals a non-analytic behavior of conditional probabilities reminiscent of a second-order phase transition. This occurs during the transition from correlation to anti-correlation in the observed probability distribution. The relation between the generalized likelihood approach and alternative methods, such as predictive likelihood and the minimum common cause entropy, is discussed. The consideration of the common cause for three observed variables (and one hidden cause) uncovers causal structures that defy representation through directed acyclic graphs with the Markov condition.