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

TL;DR

This work tackles the challenge of evaluating the G-Wishart normalising constant $\mathcal{C}_{\mathcal{G}}(\delta, D)$ for general Gaussian graphical models, where direct high-dimensional integration is intractable. By integrating Fourier analysis with chordal completions, the authors transform the problem into lower-dimensional or even one-dimensional integrals and derive explicit formulas for several graph families, including complete, complete multipartite, and graphs with minimum fill-in 1 or 2. Key contributions include a main theorem expressing $\mathcal{I}_{\mathcal{G}}(\beta,D)$ via a $\mathcal{G}^*$-based integral over $\mathbb{R}^{\tau}$, a partition-based factorisation for $D=I$ across missing edges, and specialized one-dimensional forms involving hypergeometric functions and the $U$ confluent hypergeometric function. The results enable accurate, efficient computation of normalising constants for a broad class of graphs and demonstrate practical utility on the Iris dataset, with potential to accelerate Bayesian model selection in Gaussian graphical models.

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.

Figures (11)

• Figure 1: In the graphs, solid edges represent the edges, while dashed edges represent missing edges. In the violin plots, the dots represent the estimates of the value of $\log\mathcal{C}_{\mathcal{G}}(20, I_p)$, where $\mathcal{G}$ is the corresponding graph, $p$ is the number of vertices of $\mathcal{G}$, using Monte Carlo integration Atay-Kayismw19 with $1000$ samples for 200 different seeds. They agree well with our results, represented by the horizontal lines, which have the benefit of avoiding stochastic noise and higher computational efficiency.
• Figure 2: Solid edges represent the three graphs $\mathcal{G}_1, \mathcal{G}_2, \mathcal{G}_3$ (from left to right) in Example \ref{['ex:K5K6']}. Dashed edges represent missing edges.
• Figure 4: (a) Solid edges represent the graph $\mathcal{G}(5; 2,1,1)$. Together with the dashed ones, it is the graph $\mathcal{G}^*(5; 2,1,1)$. (b) Solid edges represent the graph $\mathcal{G}$ in Section \ref{['sec:missingTriangle']}. Together with the dashed ones it is the chordal completion $\mathcal{G}^*$. The graph $\mathcal{G}$ is also the gear graph $\mathcal{G}_3$ defined in Section \ref{['sec:gear']}. (c) Solid edges represent the gear graph $\mathcal{G}_4$. Together with the dashed edges, it is the graph $\mathcal{G}_4^*$.
• Figure 5: Solid edges represent the graph $\mathcal{H}$, i.e., the cycle of length 6. Together with the dashed edges, they are chordal completions of $\mathcal{H}$. Left: $\mathcal{H}^*_1$, middle: $\mathcal{H}^*_2$, right: $\mathcal{H}^*_3$.
• Figure 6: Above: The graphs $\mathcal{G}_1$, $\mathcal{G}_2$, $\mathcal{G}_3$ (from left to right), are the non-chordal graphs for Fisher's Iris Virginica dataset. Below: Violin plots of the estimates of the values of $\log(p(Z \,|\, \mathcal{G}_j)))$, where $j = 1,2,3$, using Monte Carlo integration Atay-Kayismw19 with $10^6$ samples and for 200 different seeds. The horizontal lines represent the values obtained using our approach.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• proof : Proof of Theorem \ref{['thm:main']}
• Example 3.1
• Example 4.1
• Example 4.2
• Corollary 4.3
• proof
• Example 4.4
• Lemma 4.5