使用局部线性近似的图模型最大后验估计
Maximum a Posteriori Estimation in Graphical Models Using Local Linear Approximation
AI总结：
本文提出一种局部线性近似算法，用于解决高维高斯图模型在分层先验下的MAP估计难题，并证明其在包括图horseshoe在内的完全单调先验下收敛，所得估计稀疏且一致。
摘要：
高维高斯图模型中的稀疏结构学习是多变量统计信号处理中的一个重要问题，因为稀疏模式自然地编码了变量之间的条件独立关系。然而，在分层先验模型下，最大后验（MAP）估计具有挑战性，传统的数值优化例程或期望最大化算法难以实现。为此，我们的贡献是一种新颖的局部线性近似方案，它通过一个非常简单的计算算法绕开了这一问题。最重要的是，我们明确给出了算法保证收敛到MAP估计的条件，并证明该条件涵盖了一类广泛的完全单调先验，包括图horseshoe先验。此外，所得的MAP估计被证明在$\ell_2$范数下是稀疏且一致的。数值结果验证了所提方法的速度、可扩展性和统计性能。
英文摘要：
Sparse structure learning in high-dimensional Gaussian graphical models is an important problem in multivariate statistical signal processing; since the sparsity pattern naturally encodes the conditional independence relationship among variables. However, maximum a posteriori (MAP) estimation is challenging under hierarchical prior models, and traditional numerical optimization routines or expectation--maximization algorithms are difficult to implement. To this end, our contribution is a novel local linear approximation scheme that circumvents this issue using a very simple computational algorithm. Most importantly, the condition under which our algorithm is guaranteed to converge to the MAP estimate is explicitly stated and is shown to cover a broad class of completely monotone priors, including the graphical horseshoe. Further, the resulting MAP estimate is shown to be sparse and consistent in the $\ell_2$-norm. Numerical results validate the speed, scalability, and statistical performance of the proposed method.
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