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 implemented a Markov chain Monte Carlo algorithm and show that (i) posterior edge inclusion estimates compared to the standard technique are comparable despite searching a smaller graph space and (ii) the vector space perspective enables straightforward MCMC algorithms.
翻译:Gausian 图形模型是有条件独立结构推断多变随机变量的有用工具。 不幸的是, Bayesian 潜在图形结构的推论具有挑战性,因为以 $\ mathcal{G ⁇ n$ 的指数增长, 以 $n vertics 表示所有图表的一组。 为解决这一问题而提出的一种方法是将搜索限制在 $\ mathcal{G ⁇ n$ 的子集中。 在本文中, 我们研究的是作为循环空间矢量子空间的子集 $\ mathcal{C ⁇ n$ 作为主要例子。 我们提议在 $\ mathcal{C ⁇ n$ 之前, 以循环基元素的线性组合为基础, 并展示其理论属性。 我们使用此方法实施了 Markov 链 Monte Carlo 算法, 并显示 (i) 尽管搜索了较小的图形空间, 远端包括了比标准技术的子集, 并且(ii) 矢量空间视角使得 MC 算法更为直观 。