Testing Sparsity Assumptions in Bayesian Networks

TL;DR

This paper introduces a hypothesis test based on eigenvalues of the inverse covariance matrix to assess sparsity assumptions in Bayesian networks, aiding structure discovery.

Contribution

It extends prior theoretical results by providing asymptotic properties and a debiasing procedure for eigenvalues, enabling practical testing of maximum in-degree assumptions.

Findings

The hypothesis test accurately detects max in-degree greater than 1 in simulations.

The proposed workflow assists in selecting suitable structure discovery algorithms.

Application to psoriasis data demonstrates practical utility.

Abstract

Bayesian network (BN) structure discovery algorithms typically either make assumptions about the sparsity of the true underlying network, or are limited by computational constraints to networks with a small number of variables. While these sparsity assumptions can take various forms, frequently the assumptions focus on an upper bound for the maximum in-degree of the underlying graph $\nabla_G$. Theorem 2 in Duttweiler et. al. (2023) demonstrates that the largest eigenvalue of the normalized inverse covariance matrix ($\Omega$) of a linear BN is a lower bound for $\nabla_G$. Building on this result, this paper provides the asymptotic properties of, and a debiasing procedure for, the sample eigenvalues of $\Omega$, leading to a hypothesis test that may be used to determine if the BN has max in-degree greater than 1. A linear BN structure discovery workflow is suggested in which the investigator uses this hypothesis test to aid in selecting an appropriate structure discovery algorithm. The hypothesis test performance is evaluated through simulations and the workflow is demonstrated on data from a human psoriasis study.