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 a large class of graphs beyond chordal graphs. Furthermore, they open new possibilities for developing more efficient sampling schemes for Bayesian inference of Gaussian graphical models.

翻译：G-Wishart分布作为精度矩阵的共轭先验，是高斯图模型贝叶斯分析的核心组成部分。评估此类模型的边际似然通常需要计算高维积分以确定G-Wishart归一化常数。可分解图或弦图存在封闭解，而对一般图近年已推导出显式的形式级数展开。然而，嵌套无穷级数不便于计算，实际应用价值有限。通过借鉴随机矩阵理论与傅里叶分析方法，我们提出了新颖的精确结果，可有效评估弦图之外的大类图模型的归一化常数。此外，这些结果为开发高斯图模型贝叶斯推断的更高效采样方案开辟了新途径。