A Robustified posterior for Bayesian inference on a large number of parallel effects
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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, particularly in the tails of the distribution, can lead to erroneous estimates of the random effects, especially for the largest and most interesting effects. This paper proposes a robustified posterior distribution that eliminates the impact of a misspecified prior on one component of the standard posterior by replacing that component with an asymptotically correct form. The proposed posterior can be combined with a flexible working prior to achieve superior inference across different structures of the underlying effects of interest.
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