Qualitative Mechanism Independence

TL;DR

This paper introduces a qualitative framework for understanding how probability distributions align with causal mechanisms, extending traditional Bayesian network concepts to cyclic and functional dependencies, with deep ties to information theory.

Contribution

It defines QIM-compatibility for hypergraphs, enabling analysis of cyclic, functional, and causal structures beyond standard Bayesian networks.

Findings

QIM-compatibility captures functional dependencies.

It extends causal analysis to cyclic graphs.

Connections to information theory clarify conceptual issues.

Abstract

We define what it means for a joint probability distribution to be compatible with a set of independent causal mechanisms, at a qualitative level -- or, more precisely, with a directed hypergraph ${\mathcal{A}}$, which is the qualitative structure of a probabilistic dependency graph (PDG). When ${\mathcal{A}}$ represents a qualitative Bayesian network, QIM-compatibility with ${\mathcal{A}}$ reduces to satisfying the appropriate conditional independencies. But giving semantics to hypergraphs using QIM-compatibility lets us do much more. For one thing, we can capture functional dependencies. For another, we can capture important aspects of causality using compatibility: we can use compatibility to understand cyclic causal graphs, and to demonstrate structural compatibility, we must essentially produce a causal model. Finally, QIM-compatibility has deep connections to information theory. Applying our notion to cyclic structures helps to clarify a longstanding conceptual issue in information theory.