Annals of Statistics

Geometry of the faithfulness assumption in causal inference

Caroline Uhler, Garvesh Raskutti, Peter Bühlmann, and Bin Yu

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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.

Article information

Ann. Statist., Volume 41, Number 2 (2013), 436-463.

First available in Project Euclid: 16 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H05: Characterization and structure theory 62H20: Measures of association (correlation, canonical correlation, etc.) 14Q10: Surfaces, hypersurfaces

Causal inference PC-algorithm (strong) faithfulness conditional independence directed acyclic graph structural equation model real algebraic hypersurface Crofton’s formula algebraic statistics


Uhler, Caroline; Raskutti, Garvesh; Bühlmann, Peter; Yu, Bin. Geometry of the faithfulness assumption in causal inference. Ann. Statist. 41 (2013), no. 2, 436--463. doi:10.1214/12-AOS1080.

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