  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.
