Qualitative Mechanism Independence

TL;DR

The paper introduces Qualitative Independent-Mechanism (QIM) Compatibility, a framework that pairs joint distributions with directed hypergraphs encoding independent mechanisms. It shows QIM-compatibility generalizes conditional independence in Bayesian nets, captures functional dependencies, and yields meaningful semantics for cyclic causal structures via witnesses and generalized PSEMs. A tight causal-information-theory correspondence is established: witnesses to QIM-compatibility correspond to (generalized) randomized PSEMs and do- interventions align with causal models under independence, with information-theoretic constraints (I_muDmue muf) and a PDG-inspired scoring function (QIM muI mun muc) grounding the theory. The approach unifies causality, dependence, and information theory, extends BN reasoning to broader dependency structures, and opens questions on computation, cyclic models, and practical inference.

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.

Figures (8)

• Figure 1: $\mathbf{I}_\mu$.
• Figure 2: Illustrations of the structural deficiency $\mathit{I muD mue muf}_{\mathcal{A}}$ underneath drawn underneath their associated hypergraphs $\{ G_i\}$. Each circle represents a variable; an area in the intersection of circles $\{C_j\}$ but outside of circles $\{D_k\}$ corresponds to information that is shared between all $C_j$'s, but not in any $D_k$. Variation of a candidate distribution $\mu$ in a green area makes its qualitative fit better (according to $\mathit{I muD mue muf}{}$), while variation in a red area makes its qualitative fit worse; grey is neutral. Only the boxed structures in blue, whose $\mathit{I muD mue muf}$ can be seen as measuring distance to a particular set of (conditional) independencies, are expressible as BNs.
• Figure :
• Figure :
• Figure :

Theorems & Definitions (32)

• Definition 1
• Definition 2: QIM-compatibility
• Example 1
• Example 2
• Example 3
• Definition 3: pearl2009causality
• Definition 4
• Claim 1
• Lemma 1
• proof