We propose a novel approach to estimating the precision matrix of multivariate Gaussian data that relies on decomposing them into a low-rank and a diagonal component. Such decompositions are very popular for modeling large covariance matrices as they admit a latent factor based representation that allows easy inference. The same is however not true for precision matrices due to the lack of computationally convenient representations which restricts inference to low-to-moderate dimensional problems. We address this remarkable gap in the literature by building on a latent variable representation for such decomposition for precision matrices. The construction leads to an efficient Gibbs sampler that scales very well to high-dimensional problems far beyond the limits of the current state-of-the-art. The ability to efficiently explore the full posterior space also allows the model uncertainty to be easily assessed. The decomposition crucially additionally allows us to adapt sparsity inducing priors to shrink the insignificant entries of the precision matrix toward zero, making the approach adaptable to high-dimensional small-sample-size sparse settings. Exact zeros in the matrix encoding the underlying conditional independence graph are then determined via a novel posterior false discovery rate control procedure. A near minimax optimal posterior concentration rate for estimating precision matrices is attained by our method under mild regularity assumptions. We evaluate the method's empirical performance through synthetic experiments and illustrate its practical utility in data sets from two different application domains.
翻译:我们建议一种新颖的方法来估计多变量高斯数据的精确矩阵,该方法依靠将这些数据分解成低位和对角部分。这种分解对于模拟大型共变矩阵非常流行,因为它们承认一种基于潜在因素的表示,容易推断。然而,精确矩阵却并非如此,因为缺乏计算方便的表述,限制了对低至中位的维度问题的推论。我们利用精确矩阵变异的潜在变量代表,解决文献中的这一显著差距。构建导致高效的Gibs采样器,该采样器的大小远远超过当前状态的限度,达到高度问题的程度。有效探索整个后方空间的能力也使得模型不确定性得到容易评估。这种分解还使我们能够调整微调诱因先将精确矩阵的微分入零,使方法适应高度小型稀释环境。构建一个高效的Gibbbs采样器,其精确度非常接近当前状态的高度问题。高效的基数;高效探索整个后方空间空间的能力也使我们易于评估模型的不确定性得到容易评估。随后通过一种新的方法,通过一种最精确的精确度的模型分析方法,将一个基础的精确度的精确度的精确度的精确度模型模型的模型的模型的模型的精确度的精确度的精确度的精确度的精确度的精确度的模型的精确度的精确度的精确度的精确度用于。