Spectral Bayesian Network Theory

TL;DR

This paper introduces a spectral approach to understanding Bayesian Networks by analyzing the inverse-covariance matrix, enabling the study of global properties of the network structure rather than specific edges.

Contribution

It defines the structural hypergraph of a Bayesian Network and derives spectral bounds related to the network's maximum indegree, advancing structure learning methods.

Findings

Spectral bounds are closely related to the maximum indegree of the BN.

The structural hypergraph provides a new perspective on BN structure.

Inverse-covariance matrix analysis reveals global properties of BNs.

Abstract

A Bayesian Network (BN) is a probabilistic model that represents a set of variables using a directed acyclic graph (DAG). Current algorithms for learning BN structures from data focus on estimating the edges of a specific DAG, and often lead to many `likely' network structures. In this paper, we lay the groundwork for an approach that focuses on learning global properties of the DAG rather than exact edges. This is done by defining the structural hypergraph of a BN, which is shown to be related to the inverse-covariance matrix of the network. Spectral bounds are derived for the normalized inverse-covariance matrix, which are shown to be closely related to the maximum indegree of the associated BN.