Adjusted Bayesian inference for selected parameters

We address the problem of providing inference from a Bayesian perspective for parameters selected after viewing the data. We present a Bayesian framework for providing inference for selected parameters, based on the observation that providing Bayesian inference for selected parameters is a truncated data problem. We show that if the prior for the parameter is non-informative, or if the parameter is a 'fixed' unknown constant, then it is necessary to adjust the Bayesian inference for selection. Our second contribution is the introduction of Bayesian false discovery rate controlling methodology, which generalizes existing Bayesian false discovery rate methods that are only defined in the two-group mixture model. We illustrate our results by applying them to simulated data and data from a microarray experiment.