Gaussian graphical models are useful tools for conditional independence structure inference of multivariate random variables. Unfortunately, Bayesian inference of latent graph structures is challenging due to exponential growth of $\mathcal{G}_n$, the set of all graphs in $n$ vertices. One approach that has been proposed to tackle this problem is to limit search to subsets of $\mathcal{G}_n$. In this paper, we study subsets that are vector subspaces with the cycle space $\mathcal{C}_n$ as main example. We propose a novel prior on $\mathcal{C}_n$ based on linear combinations of cycle basis elements and present its theoretical properties. Using this prior, we implement a Markov chain Monte Carlo algorithm, and show that (i) posterior edge inclusion estimates computed with our technique are comparable to estimates from the standard technique despite searching a smaller graph space, and (ii) the vector space perspective enables straightforward implementation of MCMC algorithms.

翻译：高斯图模型是多变量随机变量条件独立性结构推断的有用工具。遗憾的是，由于$\mathcal{G}_n$（所有$n$个顶点图构成的集合）呈指数级增长，潜在图结构的贝叶斯推断极具挑战性。解决该问题的一种方法是限制搜索范围为$\mathcal{G}_n$的子集。本文以循环空间$\mathcal{C}_n$为主要实例，研究作为向量子空间的子集。我们基于循环基元素的线性组合提出了$\mathcal{C}_n$上的一种新型先验分布，并展示了其理论性质。利用该先验分布，我们实现了马尔可夫链蒙特卡洛算法，并证明：(i) 尽管搜索的图空间较小，但基于该技术的后验边包含估计与标准技术得到的估计相当；(ii) 向量空间视角使得MCMC算法的实现更加简便。