具有尖峰和平板先验的高维高斯图形模型的贝叶斯计算
Bayesian computation for high-dimensional Gaussian Graphical Models with spike-and-slab priors
AI总结：
研究高维高斯图形模型的贝叶斯计算，提出完全贝叶斯算法，包括块内吉布斯 - 梅特罗波利斯 - 黑斯廷斯算法和全局提议，可扩展到高维，给出混合边界和成本边界，扩展了精确贝叶斯推断的适用性。
摘要：
高斯图形模型被广泛用于推断依赖结构。贝叶斯方法在量化与结构学习相关的不确定性以及参数估计方面很有吸引力。然而，当变量数量很大时，计算需求限制了它们的应用，这促使使用伪贝叶斯方法。我们提出了完全贝叶斯算法，当数据生成精度矩阵稀疏时，这些算法可证明能扩展到高维，成本与最佳伪贝叶斯方法相似。首先是一种块内吉布斯-梅特罗波利斯-黑斯廷斯算法，允许按行更新精度矩阵，使用局部移动。其次是一种全局提议，能够在一行内添加或删除多条边，有助于探索多模态后验。我们获得了相对于理想吉布斯采样器的两种采样器的混合边界，在合适设置下是无维度的，并证明了这种理想吉布斯采样器是几何遍历的。我们还提供了每次迭代成本的最坏情况多项式边界，不过在实践中使用稀疏线性代数成本更低。我们的例子表明，这些方法将精确贝叶斯推断的适用性从大约100个变量扩展到大约1000个变量（相当于从大约5000条边扩展到500000条边）。
英文摘要：
Gaussian graphical models are widely used to infer dependence structures. Bayesian methods are appealing to quantify uncertainty associated with structural learning, i.e., the plausibility of conditional independence statements given the data, and parameter estimates. However, computational demands have limited their application when the number of variables is large, which prompted the use of pseudo-Bayesian approaches. We propose fully Bayesian algorithms that provably scale to high dimensions when the data-generating precision matrix is sparse, at a similar cost to the best pseudo-Bayesian methods. First, a Metropolis-Hastings-within-Block-Gibbs algorithm that allows row-wise updates of the precision matrix, using local moves. Second, a global proposal that enables adding or removing multiple edges within a row, which can help explore multi-modal posteriors. We obtain mixing bounds for both samplers relative to an ideal Gibbs sampler that are dimension-free under suitable settings, and prove that this ideal Gibbs sampler is geometrically ergodic. We also provide worst-case polynomial bounds on per-iteration costs, though in practice the cost is lower by using sparse linear algebra. Our examples show that the methods extend the applicability of exact Bayesian inference from roughly 100 to roughly 1000 variables (equivalently, from 5,000 edges to 500,000 edges).
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