Posterior convergence rates for estimating large precision matrices using graphical models
Posterior convergence rates for estimating large precision matrices using graphical models
复制标题
使用图模型估计大型精度矩阵的后验收敛率
DOI：
10.1214/14-ejs945
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发表时间：
2013
影响因子：
1.1
通讯作者：
S. Ghosal
中科院分区：
文献类型：
--
作者：
Sayantan Banerjee;S. Ghosal
我们考虑贝叶斯估计的$p\times p$精度矩阵，当$p$可以远远大于可用的样本量$n$。众所周知，在这种超高维情况下的一致估计需要正则化，例如条带化、锥形化或阈值化。我们考虑了带状结构的模型，并通过高斯图形模型，其中只有当两个顶点是在给定的距离内的边缘是存在的带状精度矩阵上的先验分布。在适当选择图的阶数的情况下，得到了一类精度矩阵上基于图模型的后验分布和Bayes估计在L_{\infty}$-算子范数下的一致收敛速度，即使真精度矩阵可能不具有带状结构.沿着证明的过程中，我们还计算了极大似然估计（MLE）在相同条件下的收敛速度，这是独立的兴趣。基于图模型的极大似然估计和贝叶斯估计是自动正定的，这是文献中其他估计所不具备的一个理想性质。我们还进行了模拟研究，以比较有限样本的贝叶斯估计和极大似然估计的性能的基础上的图形模型，通过使用Cholesky分解的精度矩阵。最后讨论了利用边际似然函数选择图模型阶数的一种实用方法。
We consider Bayesian estimation of a $p\times p$ precision matrix, when $p$ can be much larger than the available sample size $n$. It is well known that consistent estimation in such ultra-high dimensional situations requires regularization such as banding, tapering or thresholding. We consider a banding structure in the model and induce a prior distribution on a banded precision matrix through a Gaussian graphical model, where an edge is present only when two vertices are within a given distance. For a proper choice of the order of graph, we obtain the convergence rate of the posterior distribution and Bayes estimators based on the graphical model in the $L_{\infty}$-operator norm uniformly over a class of precision matrices, even if the true precision matrix may not have a banded structure. Along the way to the proof, we also compute the convergence rate of the maximum likelihood estimator (MLE) under the same set of condition, which is of independent interest. The graphical model based MLE and Bayes estimators are automatically positive definite, which is a desirable property not possessed by some other estimators in the literature. We also conduct a simulation study to compare finite sample performance of the Bayes estimators and the MLE based on the graphical model with that obtained by using a Cholesky decomposition of the precision matrix. Finally, we discuss a practical method of choosing the order of the graphical model using the marginal likelihood function.
影响因子：
3.7
作者：
Dobra A;Lenkoski A;Rodriguez A
通讯作者：
Rodriguez A