  Linear shrinkage estimators of a covariance matrix --- defined by a weighted
average of the sample covariance matrix and a pre-specified shrinkage target
matrix --- are popular when analysing high-throughput molecular data. However,
their performance strongly relies on an appropriate choice of target matrix.
This paper introduces a more flexible class of linear shrinkage estimators that
can accommodate multiple shrinkage target matrices, directly accounting for the
uncertainty regarding the target choice. This is done within a conjugate
Bayesian framework, which is computationally efficient. Using both simulated
and real data, we show that the proposed estimator is less sensitive to target
misspecification and can outperform state-of-the-art (nonparametric)
single-target linear shrinkage estimators. Using protein expression data from
The Cancer Proteome Atlas we illustrate how multiple sources of prior
information (obtained from more than 30 different cancer types) can be
incorporated into the proposed multi-target linear shrinkage estimator. In
particular, it is shown that the target-specific weights can provide insights
into the differences and similarities between cancer types. Software for the
method is freely available as an R-package at http://github.com/HGray384/TAS.
