Bayesian methods constitute a popular approach for estimating the conditional independence structure in Gaussian graphical models, since they can quantify the uncertainty through the posterior distribution. Inference in this framework is typically carried out with Markov chain Monte Carlo (MCMC). However, the most widely used choice of prior distribution for the precision matrix, the so called G-Wishart distribution, suffers from an intractable normalizing constant, which gives rise to the problem of double intractability in the updating steps of the MCMC algorithm. In this article, we propose a new class of prior distributions for the precision matrix, termed ST priors, that allow for the construction of MCMC algorithms that do not suffer from double intractability issues. A realization from an ST prior distribution is obtained by applying a sparsifying transform on a matrix from a distribution with support in the set of all positive definite matrices. We carefully present the theory behind the construction of our proposed class of priors and also perform some numerical experiments, where we apply our methods on a human gene expression dataset. The results suggest that our proposed MCMC algorithm is able to converge and achieve acceptable mixing when applied on the real data.

翻译：贝叶斯方法因其能够通过后验分布量化不确定性，已成为估计高斯图模型中条件独立结构的常用方法。在此框架下，推断通常通过马尔可夫链蒙特卡罗（MCMC）方法实现。然而，当前精度矩阵最广泛使用的先验分布——即所谓的G-Wishart分布——存在归一化常数难以处理的问题，这导致MCMC算法的更新步骤出现双重难解性困境。本文提出了一类新的精度矩阵先验分布，称为ST先验，其能够构建不受双重难解性问题影响的MCMC算法。ST先验分布的实现是通过对来自支撑集为所有正定矩阵集合的分布中的矩阵施加稀疏化变换而获得的。我们详细阐述了所提先验分布类别构建背后的理论，并进行了数值实验，将我们的方法应用于人类基因表达数据集。结果表明，所提出的MCMC算法在真实数据上能够收敛并获得可接受的混合效果。