Hypersurfaces and their singularities in partial correlation testing

TL;DR

This paper develops an asymptotic theory for calculating volumes of parameter regions in Gaussian graphical models defined by partial correlation bounds, using algebraic geometry techniques to analyze singularities.

Contribution

It introduces a novel algebraic geometric approach to study correlation hypersurfaces and their singularities in Gaussian models, with applications to causal inference and structure learning.

Findings

Computed volumes of parameter regions for various graph types

Analyzed singular loci of correlation hypersurfaces

Applied theory to causal inference scenarios

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.