Classifying Causal Structures: Ascertaining Inequality Constraints

Shashaank Khanna, Marina Maciel Ansanelli, Mathew F. Pusey, Elie Wolfe

Keywords: Inequalities, Causal Inference, Latent Variables, Latent Nodes, Algebraic, Inequality Constraints, Graphical Models, mDAGs

TL;DR: Some causal structures imply inequality constraints, others do not, we use everything we can to tell the difference.

Abstract: The causal relations between a set of variables, some observed and some latent, can induce both equality constraints (typically conditional independencies) as well as inequality constraints (Instrumental and Bell inequalities being prototypical examples) on their compatible distribution over the observed variables. Enumerating a causal structure’s implied inequality constraints is generally far more difficult than enumerating its equalities. Furthermore, only inequality constraints ever admit violation by quantum correlations. For both those reasons, it is important to classify causal scenarios into those which impose inequality constraints versus those which do not. Here we develop methods for detecting such scenarios by appealing to d-separation, e-separation, and incompatible supports. Many (perhaps all?) scenarios with exclusively equality constraints can be detected via a condition articulated by Henson, Lal and Pusey (HLP). Considering all scenarios with up to 4 observed variables, which number in the thousands, we are able to resolve all but three causal scenarios, providing evidence that the HLP condition is, in fact, exhaustive.

Submission Number: 6