Causal Models with Constraints

Sander Beckers, Joseph Halpern, Christopher Hitchcock

Keywords: Causality, Constraints, Interventions, Abstractions

TL;DR: We generalize causal models to allow for non-causal relations between variables in addition to the causal ones.

Abstract: Causal models have proven extremely useful in offering formal representations of causal relationships between a set of variables. Yet in many situations, there are non-causal relationships among variables. For example, we may want variables $LDL$, $HDL$, and $TOT$ that represent the level of low-density lipoprotein cholesterol, the level of high-density lipoprotein cholesterol, and total cholesterol level, with the relation $LDL+HDL = TOT$. This cannot be done in standard causal models, because we can intervene simultaneously on all three variables. The goal of this paper is to extend standard causal models to allow for constraints on settings of variables. Although the extension is relatively straightforward, to make it useful we have to define a new intervention operation that disconnects a variable from a causal equation. We give examples showing the usefulness of this extension, and provide a sound and complete axiomatization for causal models with constraints.