高斯精度矩阵的惩罚似然估计的存在性及可分离先验分布的后验正则性
Existence of penalised likelihood estimates and posterior propriety of separable prior distributions for Gaussian precision matrices
AI总结：
本文分析高维下高斯精度矩阵惩罚似然估计的存在性，给出确保其存在的惩罚函数条件，并拓展至贝叶斯场景，给出确保后验正则的可分离先验分布条件。
摘要：
惩罚似然常被用于高斯精度矩阵的稀疏估计。在矩阵维度大于样本量的高维场景中，样本协方差矩阵$S$非满秩，精度矩阵的最大似然估计不存在。部分惩罚似然估计（例如graphical lasso）的额外优势在于，即便在这类高维场景中也能存在。本文对正半定$S$的惩罚似然估计的存在性展开了全面分析，针对对角与非对角惩罚函数给出了确保估计存在的特定尾部条件；该分析还被拓展至贝叶斯场景，给出了确保所得后验分布正则的可分离先验分布条件。
英文摘要：
Penalised likelihoods are often used for sparse estimation of a Gaussian precision matrix. In high dimensional settings where the matrix dimension is larger than the sample size, the sample covariance matrix $S$ is not of full rank and the maximum likelihood estimate of the precision matrix does not exist. An additional advantage of some penalised likelihood estimates, for example the graphical lasso, is that it can exist even in such high dimensional settings. This paper gives a thorough analysis of the existence of penalised likelihood estimates for positive semidefinite $S$. Specific tail conditions are provided on the diagonal and off-diagonal penalty functions that ensure existence of the estimate. This is also extended to the Bayesian setting where conditions on separable prior distributions are provided that ensure the resulting posterior distribution is proper.
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