The Bayesian Covariance Lasso.
The Bayesian Covariance Lasso.
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DOI：
10.4310/sii.2013.v6.n2.a8
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发表时间：
2013-04-01
影响因子：
0.8
通讯作者：
Ibrahim JG
中科院分区：
文献类型：
--
作者：
Khondker ZS;Zhu H;Chu H;Lin W;Ibrahim JG
近年来，稀疏协方差矩阵及其逆正定性约束的估计引起了广泛的关注。样本量 (n) 小于维度 (d) 的大量高维数据需要收缩估计方法，因为在这种情况下最大似然估计量不是正定的。此外，当n大于d但又不够大时，收缩估计比最大似然更稳定，因为它减少了精度矩阵的条件数。频率主义方法利用了惩罚似然方法，而贝叶斯方法则依赖于矩阵分解或 Wishart 先验来进行收缩。在本文中，我们提出了一种新方法，称为贝叶斯协方差套索（BCLASSO），用于精度（协方差）矩阵的收缩估计。我们将导致流行的频率论惩罚的精度矩阵的一类先验视为特殊情况，开发精度矩阵的贝叶斯估计器，并提出一种不预先计算正定性边界的有效采样方案。所提出的方法是排列不变的，并且对非满秩数据同时执行收缩和估计。仿真表明，所提出的 BCLASSO 的性能与非满秩数据的频率主义方法类似。
Estimation of sparse covariance matrices and their inverse subject to positive definiteness constraints has drawn a lot of attention in recent years. The abundance of high-dimensional data, where the sample size (n) is less than the dimension (d), requires shrinkage estimation methods since the maximum likelihood estimator is not positive definite in this case. Furthermore, when n is larger than d but not sufficiently larger, shrinkage estimation is more stable than maximum likelihood as it reduces the condition number of the precision matrix. Frequentist methods have utilized penalized likelihood methods, whereas Bayesian approaches rely on matrix decompositions or Wishart priors for shrinkage. In this paper we propose a new method, called the Bayesian Covariance Lasso (BCLASSO), for the shrinkage estimation of a precision (covariance) matrix. We consider a class of priors for the precision matrix that leads to the popular frequentist penalties as special cases, develop a Bayes estimator for the precision matrix, and propose an efficient sampling scheme that does not precalculate boundaries for positive definiteness. The proposed method is permutation invariant and performs shrinkage and estimation simultaneously for non-full rank data. Simulations show that the proposed BCLASSO performs similarly as frequentist methods for non-full rank data.