On the Equivalence of Gaussian Graphical Models Defined on Complete Bipartite Graphs

TL;DR

This paper explores the relationship between two Gaussian graphical models on complete bipartite graphs, revealing their determinants are proportionally equal and introducing an equivalence concept that simplifies calculations and yields closed-form solutions.

Contribution

It establishes a notion of equivalence between two models, reducing computational complexity and enabling explicit determinant formulas in special cases.

Findings

Determinants of the models' precision matrices are proportional.

Equivalence simplifies determinant computation.

Closed-form expressions are derived for specific cases.

Abstract

This paper introduces two Gaussian graphical models defined on complete bipartite graphs. We show that the determinants of the precision matrices associated with the models are equal up to scale, where the scale factor only depends on model parameters. In this context, we will introduce a notion of ``equivalence" between the two Gaussian graphical models. This equivalence has two key applications: first, it can significantly reduce the complexity of computing the exact value of the determinant, and second, it enables the derivation of closed-form expressions for the determinants in certain special cases.