A new way to evaluate G-Wishart normalising constants via Fourier analysis

TL;DR

This paper introduces a novel Fourier analysis-based method to accurately compute G-Wishart normalising constants for Gaussian graphical models, extending applicability beyond chordal graphs.

Contribution

It presents a new Fourier analysis approach that enables practical numerical evaluation of G-Wishart normalising constants for complex graph structures.

Findings

Provides exact formulas suitable for numerical computation

Extends evaluation methods to non-chordal graphs

Improves computational efficiency over previous series expansions

Abstract

The G-Wishart distribution is an essential component for the Bayesian analysis of Gaussian graphical models as the conjugate prior for the precision matrix. Evaluating the marginal likelihood of such models usually requires computing high-dimensional integrals to determine the G-Wishart normalising constant. Closed-form results are known for decomposable or chordal graphs, while an explicit representation as a formal series expansion has been derived recently for general graphs. The nested infinite sums, however, do not lend themselves to computation, remaining of limited practical value. Borrowing techniques from random matrix theory and Fourier analysis, we provide novel exact results well suited to the numerical evaluation of the normalising constant for classes of graphs beyond chordal graphs.