  Estimating the sharing of genetic effects across different conditions is
important to many statistical analyses of genomic data. The patterns of sharing
arising from these data are often highly heterogeneous. To flexibly model these
heterogeneous sharing patterns, Urbut et al. (2019) proposed the multivariate
adaptive shrinkage (MASH) method to jointly analyze genetic effects across
multiple conditions. However, multivariate analyses using MASH (as well as
other multivariate analyses) require good estimates of the sharing patterns,
and estimating these patterns efficiently and accurately remains challenging.
Here we describe new empirical Bayes methods that provide improvements in speed
and accuracy over existing methods. The two key ideas are: (1) adaptive
regularization to improve accuracy in settings with many conditions; (2)
improving the speed of the model fitting algorithms by exploiting analytical
results on covariance estimation. In simulations, we show that the new methods
provide better model fits, better out-of-sample performance, and improved power
and accuracy in detecting the true underlying signals. In an analysis of eQTLs
in 49 human tissues, our new analysis pipeline achieves better model fits and
better out-of-sample performance than the existing MASH analysis pipeline. We
have implemented the new methods, which we call ``Ultimate Deconvolution'', in
an R package, udr, available on GitHub.
