Open Access
Translator Disclaimer
April 2013 Geometry of the faithfulness assumption in causal inference
Caroline Uhler, Garvesh Raskutti, Peter Bühlmann, Bin Yu
Ann. Statist. 41(2): 436-463 (April 2013). DOI: 10.1214/12-AOS1080


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.


Download Citation

Caroline Uhler. Garvesh Raskutti. Peter Bühlmann. Bin Yu. "Geometry of the faithfulness assumption in causal inference." Ann. Statist. 41 (2) 436 - 463, April 2013.


Published: April 2013
First available in Project Euclid: 16 April 2013

zbMATH: 1267.62068
MathSciNet: MR3099109
Digital Object Identifier: 10.1214/12-AOS1080

Primary: 14Q10 , 62H05 , 62H20

Keywords: (strong) faithfulness , Algebraic statistics , Causal inference , Conditional independence , Crofton’s formula , Directed acyclic graph , PC-algorithm , real algebraic hypersurface , structural equation model

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 2 • April 2013
Back to Top