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Geometry of the faithfulness assumption in causal inference. (English) Zbl 1267.62068
Summary: 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.

62H05 Characterization and structure theory for multivariate probability distributions; copulas
05C20 Directed graphs (digraphs), tournaments
65C60 Computational problems in statistics (MSC2010)
62H20 Measures of association (correlation, canonical correlation, etc.)
05C90 Applications of graph theory
TETRAD; pcalg
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