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.

中文翻译：

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贝叶斯协方差套索

近年来，稀疏协方差矩阵的估计及其受正定性约束的逆矩阵引起了很多关注。大量的高维数据，其中样本大小 (n) 小于维度 (d)，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。频率论方法使用惩罚似然方法，而贝叶斯方法依靠矩阵分解或 Wishart 先验来进行收缩。在本文中，我们提出了一种称为贝叶斯协方差套索 (BCLASSO) 的新方法，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，开发了精度矩阵的贝叶斯估计量，并提出了一种不预先计算正定边界的有效采样方案。所提出的方法是置换不变的，并且对非满秩数据同时进行收缩和估计。模拟表明，所提出的 BCLASSO 与非满秩数据的频率论方法相似。所提出的方法是置换不变的，并且对非满秩数据同时进行收缩和估计。模拟表明，所提出的 BCLASSO 与非满秩数据的频率论方法相似。所提出的方法是置换不变的，并且对非满秩数据同时进行收缩和估计。模拟表明，所提出的 BCLASSO 与非满秩数据的频率论方法相似。