Partial correlation hypersurfaces in Gaussian graphical models

We derive a combinatorial sufficient condition for a partial correlation hypersurface in the parameter space of a directed Gaussian graphical model to be nonsingular, and speculate on whether this condition can be used in algorithms for learning the graph. Since the condition is fulfilled in the case of a complete DAG on any number of vertices, the result implies an affirmative answer to a question raised by Lin-Uhler-Sturmfels-B\"uhlmann.Comment: 9 pages, 5 figures, added Example 13, some minor further edit

Geometry of the faithfulness assumption in causal inference

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.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1080 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Direct Estimation of Differences in Causal Graphs

We consider the problem of estimating the differences between two causal directed acyclic graph (DAG) models with a shared topological order given i.i.d. samples from each model. This is of interest for example in genomics, where changes in the structure or edge weights of the underlying causal graphs reflect alterations in the gene regulatory networks. We here provide the first provably consistent method for directly estimating the differences in a pair of causal DAGs without separately learning two possibly large and dense DAG models and computing their difference. Our two-step algorithm first uses invariance tests between regression coefficients of the two data sets to estimate the skeleton of the difference graph and then orients some of the edges using invariance tests between regression residual variances. We demonstrate the properties of our method through a simulation study and apply it to the analysis of gene expression data from ovarian cancer and during T-cell activation

Hypersurfaces and their singularities in partial correlation testing

An asymptotic theory is developed for computing volumes of regions in the parameter space of a directed Gaussian graphical model that are obtained by bounding partial correlations. We study these volumes using the method of real log canonical thresholds from algebraic geometry. Our analysis involves the computation of the singular loci of correlation hypersurfaces. Statistical applications include the strong-faithfulness assumption for the PC algorithm and the quantification of confounder bias in causal inference. A detailed analysis is presented for trees, bow ties, tripartite graphs, and complete graphs