A major problem in numerical weather prediction (NWP) is the estimation of high-dimensional covariance matrices from a small number of samples. Maximum likelihood estimators cannot provide reliable estimates when the overall dimension is much larger than the number of samples. Fortunately, NWP practitioners have found ingenious ways to boost the accuracy of their covariance estimators by leveraging the assumption that the correlations decay with spatial distance. In this work, Bayesian statistics is used to provide a new justification and analysis of the practical NWP covariance estimators. The Bayesian framework involves manipulating distributions over symmetric positive definite matrices, and it leads to two main findings: (i) the commonly used "hybrid estimator" for the covariance matrix has a naturally Bayesian interpretation; (ii) the very commonly used "Schur product estimator" is not Bayesian, but it can be studied and understood within the Bayesian framework. As practical implications, the Bayesian framework shows how to reduce the amount of tuning required for covariance estimation, and it suggests that efficient covariance estimation should be rooted in understanding and penalizing conditional correlations, rather than correlations.

翻译：数字天气预测(NWP)的一个主要问题是从少数样本中估算高维共变矩阵。当总体尺寸大大大于样本数量时,最大概率估计者无法提供可靠的估计。幸运的是,NWP从业人员发现一些奇特的方法,利用相关关系与空间距离衰减的假设,提高了共变估计者的准确性。在这项工作中,巴耶斯统计用于为实际的NWP共变估计者提供新的理由和分析。贝叶斯框架涉及对正对正确定矩阵进行操纵分布,它导致两个主要结论:(一)常使用的共变矩阵“混合估计者”具有自然的巴耶斯解释;(二)常用的“Schur产品估计者”不是拜耶斯人,但可在Bayesian框架内加以研究和理解。作为实际影响,Bayesian框架表明如何减少对对正对正确定矩阵的调整量,并导致两个主要结论:(一)常用“混合估计者”具有自然的巴耶斯解释;(二)通常使用的“Schur产品估计者”不是拜斯人,但可在Bayesian框架内加以研究和理解。作为实际影响,Bayesian框架表明如何减少对调度估计所需的数量,它,它意味着有效的相互之间应植根基。