Hypersurfaces and Their Singularities in Partial Correlation Testing |
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Authors: | Shaowei Lin Caroline Uhler Bernd Sturmfels Peter Bühlmann |
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Institution: | 1. Institute for Infocomm Research, A*STAR, Singapore, Singapore 3. Institute of Science and Technology Austria, Klosterneuburg, Austria 2. Department of Mathematics, University of California Berkeley, Berkeley, CA, USA 4. Seminar for Statistics, ETH Zürich, Zürich, Switzerland
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Abstract: | 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. |
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