Geometry of the faithfulness assumption in causal inference

TL;DR

This paper investigates the geometric properties of the strong-faithfulness assumption in causal inference, revealing its limitations and implications for algorithms like PC, especially in high-dimensional Gaussian settings.

Contribution

It provides bounds on the measure of strong-faithful distributions and analyzes their impact on the reliability of causal inference algorithms.

Findings

Strong-faithful distributions can have nonzero measure, often larger than expected.

Fundamental limitations are identified for the PC-algorithm under strong-faithfulness.

Results suggest challenges for high-dimensional causal inference using partial correlation tests.

Abstract

Many algorithms for inferring causality rely heavily on the faithfulness assumption. The main justification for imposing this assumption is that the set of unfaithful distributions has Lebesgue measure zero, since it can be seen as a collection of hypersurfaces in a hypercube. However, due to sampling error the faithfulness condition alone is not sufficient for statistical estimation, and strong-faithfulness has been proposed and assumed to achieve uniform or high-dimensional consistency. In contrast to the plain faithfulness assumption, the set of distributions that is not strong-faithful has nonzero Lebesgue measure and in fact, can be surprisingly large as we show in this paper. We study the strong-faithfulness condition from a geometric and combinatorial point of view and give upper and lower bounds on the Lebesgue measure of strong-faithful distributions for various classes of directed acyclic graphs. Our results imply fundamental limitations for the PC-algorithm and potentially also for other algorithms based on partial correlation testing in the Gaussian case.