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Hypersurfaces and Their Singularities in Partial Correlation Testing

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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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Acknowledgments

This project began at the workshop Algebraic Statistics in the Alleghenies, which took place at Pennsylvania State University in June 2012. We are grateful to the organizers for a very inspiring meeting. We thank Thomas Richardson for pointing out the connection to the problem of bias reduction in causal inference. We also thank all reviewers for thoughtful comments and one of the reviewers in particular for numerous detailed comments and for spotting a mistake in a proof. This work was supported in part by the US National Science Foundation (DMS-0968882) and the Defense Advanced Research Projects Agency (DARPA) Deep Learning program (FA8650-10-C-7020).

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Authors and Affiliations

  1. Institute for Infocomm Research, A*STAR, Singapore, Singapore

    Shaowei Lin

  2. Department of Mathematics, University of California Berkeley, Berkeley, CA, USA

    Bernd Sturmfels

  3. Institute of Science and Technology Austria, Klosterneuburg, Austria

    Caroline Uhler

  4. Seminar for Statistics, ETH Zürich, Zürich, Switzerland

    Peter Bühlmann

Authors
  1. Shaowei Lin
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  2. Caroline Uhler
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  3. Bernd Sturmfels
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  4. Peter Bühlmann
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Corresponding author

Correspondence to Caroline Uhler.

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Communicated by Michael Todd.

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Lin, S., Uhler, C., Sturmfels, B. et al. Hypersurfaces and Their Singularities in Partial Correlation Testing. Found Comput Math 14, 1079–1116 (2014). https://doi.org/10.1007/s10208-014-9205-0

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  • DOI: https://doi.org/10.1007/s10208-014-9205-0

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