  Many modern experiments, such as microarray gene expression and genome-wide
association studies, present the problem of estimating a large number of
parallel effects. Bayesian inference is a popular approach for analyzing such
data by modeling the large number of unknown parameters as random effects from
a common prior distribution. However, misspecification of the prior
distribution can lead to erroneous estimates of the random effects, especially
for the largest and most interesting effects. This paper has two aims. First,
we propose a robustified posterior distribution for a parametric Bayesian
hierarchical model that can substantially reduce the impact of a misspecified
prior. Second, we conduct a systematic comparison of the standard parametric
posterior, the proposed robustified parametric posterior, and a nonparametric
Bayesian posterior which uses a Dirichlet process mixture prior. The proposed
robustifed posterior when combined with a flexible parametric prior can be a
superior alternative to nonparametric Bayesian methods.
