Markov Boundary Discovery with Ridge Regularized Linear Models

TL;DR

This paper demonstrates that a modified form of ridge regularized linear models can effectively identify parts of the Markov boundary, offering a promising approach for causal variable discovery in high-dimensional data.

Contribution

The paper introduces a modified RRLM approach that can approximate the Markov boundary under certain assumptions, extending its applicability to nonlinear relationships and non-unique solutions.

Findings

Modified RRLMs can identify subsets of the Markov boundary.

The approach provides worst-case bounds on solution space.

Experimental results show competitive performance in gene expression data.

Abstract

Ridge regularized linear models (RRLMs), such as ridge regression and the SVM, are a popular group of methods that are used in conjunction with coefficient hypothesis testing to discover explanatory variables with a significant multivariate association to a response. However, many investigators are reluctant to draw causal interpretations of the selected variables due to the incomplete knowledge of the capabilities of RRLMs in causal inference. Under reasonable assumptions, we show that a modified form of RRLMs can get very close to identifying a subset of the Markov boundary by providing a worst-case bound on the space of possible solutions. The results hold for any convex loss, even when the underlying functional relationship is nonlinear, and the solution is not unique. Our approach combines ideas in Markov boundary and sufficient dimension reduction theory. Experimental results show that the modified RRLMs are competitive against state-of-the-art algorithms in discovering part of the Markov boundary from gene expression data.