We consider the problem of fully Bayesian posterior estimation and uncertainty quantification in undirected Gaussian graphical models via Markov chain Monte Carlo (MCMC) under recently-developed element-wise graphical priors, such as the graphical horseshoe. Unlike the conjugate Wishart family, these priors are non-conjugate; but have the advantage that they naturally allow one to encode a prior belief of sparsity in the off-diagonal elements of the precision matrix, without imposing a structure on the entire matrix. Unfortunately, for a graph with $p$ nodes and with $n$ samples, the state-of-the-art MCMC approaches for the element-wise priors achieve a per iteration complexity of ${O}(p^4),$ which is prohibitive when $p\gg n$. In this regime, we develop a suitably reparameterized MCMC with per iteration complexity of ${O}(p^3)$, providing a one order of magnitude improvement, and consequently bringing the per iteration computational cost at par with the conjugate Wishart family, which is also ${O}(p^3)$ due to a use of the classical Bartlett decomposition, but this decomposition does not apply outside the Wishart family. Importantly, the proposed benefit is obtained solely due to our reparameterization in an MCMC scheme targeting the true posterior, that reverses the recently developed telescoping block decomposition of Bhadra et al. (2024), in a suitable sense. There is no variational or any other approximate Bayesian computation scheme considered in this paper that compromises targeting the true posterior. Simulations and the analysis of a breast cancer data set confirm both the correctness and better algorithmic scaling of the proposed reverse telescoping sampler.

翻译：我们考虑在最近发展的元素图先验（如图形马蹄形先验）下，通过马尔可夫链蒙特卡洛（MCMC）对无向高斯图模型进行全贝叶斯后验估计和不确定性量化的问题。与共轭的Wishart族不同，这些先验是非共轭的，但具有天然优势：允许在精度矩阵的非对角元素中编码稀疏性的先验信念，而无需对整个矩阵施加结构。然而，对于一个包含$p$个节点和$n$个样本的图，现有针对元素先验的最优MCMC方法每轮迭代复杂度达到${O}(p^4)$，这在$p\gg n$的情况下是难以承受的。针对这一情形，我们开发了一种经适当重参数化的MCMC方法，其每轮迭代复杂度为${O}(p^3)$，实现了一个数量级的提升，并因此使每轮计算成本与共轭Wishart族相当——后者由于使用了经典Bartlett分解也达到${O}(p^3)$复杂度，但该分解不适用于Wishart族以外的情形。重要的是，所提出的优势完全源于我们在MCMC方案中针对真实后验的重参数化，该方案在适当意义上逆转了Bhadra等人（2024）最近发展的伸缩块分解。本文未采用任何变分或其他近似贝叶斯计算方法，从而不牺牲对真实后验的逼近。模拟实验和对乳腺癌数据集的分析验证了所提出的反向伸缩采样器的正确性和更优的算法扩展性。