On Different Notions of Redundancy in Conditional-Independence-Based Discovery of Graphical Models

TL;DR

This paper investigates how conditional-independence-based graph discovery can leverage redundant CI-statements to detect and correct errors. It distinguishes purely graphical redundancy—statements that follow from graphical structure but not from probabilistic laws—from Graphoid-redundancy, which follows from semi-Graphoid axioms and holds for all distributions, thereby offering limited error-detection value. By analyzing spanning trees and DAGs, the authors derive criteria and sufficient conditions under which purely graphical redundancy can reveal mis-specifications and even correct certain errors, while warning that some redundant statements may mislead if applied indiscriminately. Empirical results show that purely graphically redundant CI-tests can indicate more errors than Graphoid-redundant tests and can improve certain error-correction strategies (e.g., Minimum Markov-Distance for trees), though corrections in DAGs are more constrained and require additional assumptions. Overall, the work advances a framework for evaluating and enhancing CI-based graph discovery by accounting for the intertwined roles of graphical and probabilistic constraints, with implications for robust causal structure learning and model validation.

Abstract

Conditional-independence-based discovery uses statistical tests to identify a graphical model that represents the independence structure of variables in a dataset. These tests, however, can be unreliable, and algorithms are sensitive to errors and violated assumptions. Often, there are tests that were not used in the construction of the graph. In this work, we show that these redundant tests have the potential to detect or sometimes correct errors in the learned model. But we further show that not all tests contain this additional information and that such redundant tests have to be applied with care. Precisely, we argue that the conditional (in)dependence statements that hold for every probability distribution are unlikely to detect and correct errors - in contrast to those that follow only from graphical assumptions.

Figures (16)

• Figure 1: The marginal independence tests identify the faithful DAG (with $Y$ as a single variable). But the collider structure $X_1-Y-X_2$ implies $X_1\not\mathrel{\hbox{$\perp$}\mkern2mu{\perp}} X_2 \mid Y$ for all faithful distributions. This does not hold for every distribution. On the contrary, given the marginal tests we have, e.g., $X_1\not \mathrel{\hbox{$\perp$}\mkern2mu{\perp}} Y\mid X_2$ for all distributions.
• Figure 2: Hierarchy of \ref{['def:graphical_redundancy', 'def:graphoid_redundancy', 'def:prob_redundant']}. We argue to use graphically but not probabilistically redundant CIs.
• Figure 3: In this graph, every dependence along more than one edge is purely graphically redundant given the CI-statements from \ref{['prop:markov_network_markovian']}.
• Figure 4: A false test $Y\not\mathrel{\hbox{$\perp$}\mkern2mu{\perp}} Z$ may lead to the conclusion that the true graph is $X\to Y\to Z$. If this were the only error, the test $Y\mathrel{\hbox{$\perp$}\mkern2mu{\perp}} Z\mid X$ would correct that. But $Y\not\mathrel{\hbox{$\perp$}\mkern2mu{\perp}} Z\mid X$ follows via Graphoid axioms from the marginal tests.
• Figure 5: Experimental results

Theorems & Definitions (38)

• definition 1: graphical redundancy
• proposition 1: continuity of a Graphoid
• definition 2: probabilistic redundancy
• definition 3: Graphoid-redundancy
• definition 4: purely graphical redundancy
• corollary 1: verma1990causal Thm. 2
• corollary 2: geiger1993logical Thm. 12
• definition 5: coupling over nodes
• proposition 2: sufficient criterion
• corollary 3: transitive dependence