使用反向折叠块分解实现高斯图模型后验采样的时间复杂度降低
An Order of Magnitude Time Complexity Reduction for Gaussian Graphical Model Posterior Sampling Using a Reverse Telescoping Block Decomposition
AI总结：
本文提出一种反向折叠块分解的MCMC方法，将高斯图模型后验估计的时间复杂度从O(p⁴)降低到O(p³)，从而在大规模数据下提升了计算效率。
摘要：
我们考虑通过马尔可夫链蒙特卡洛（MCMC）方法在无向高斯图模型中进行完全贝叶斯后验估计和不确定性量化的问题，使用 recently-developed element-wise graphical priors，如图形 horseshoe。不同于共轭 Wishart 家族，这些先验是非共轭的；但具有优势，允许自然地编码对精度矩阵对角线外元素稀疏性的先验信念，而无需对整个矩阵施加结构。不幸的是，对于具有 p 个节点和 n 个样本的图，针对 element-wise 先验的最先进 MCMC 方法每迭代的复杂度为 O(p⁴)，当 p ≫ n 时是不可行的。在该 regime 中，我们开发了一种重新参数化的 MCMC 方法，每迭代的复杂度为 O(p³)，提供了一种数量级的改进，从而将每迭代的计算成本与共轭 Wishart 家族相媲美，后者也是由于使用了经典的 Bartlett 分解，但这种分解不适用于 Wishart 家族之外。重要的是，所提出的益处仅由于我们 MCMC 方案中针对真实后验的重新参数化，反向了 recently developed telescoping block decomposition of Bhadra et al. (2024)，以某种意义。本文没有考虑变分或任何其他近似贝叶斯计算方案，这些方案会妥协于真实后验。模拟和乳腺癌数据集的分析证实了所提出反向折叠采样器的正确性和更好的算法扩展性。
英文摘要：
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.
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