Adjusted Bayesian inference for selected parameters

We address the problem of providing inference for parameters selected after viewing the data from a Bayesian perspective. A frequentist solution to this problem is constructing False Coverage-statement Rate adjusted confidence intervals for the subset of selected parameters. We illustrate the limitations of the frequentist solution. We argue that if the parameter is elicited a non-informative prior, or if it is a ``fixed'' effect that is generated before selection is applied, then it is necessary to adjust the Bayesian inference for selection. Our main contribution is a Bayesian framework for providing inference for selected parameters, based on the observation that from a Bayesian perspective providing inference for a selected parameter is a truncation problem. Our second contribution is the introduction of Bayesian FDR controlling methodology, that generalizes existing Bayesian FDR methods to the case of non-dichotomous parameters. We illustrate our results by applying them to simulated data and data from a microarray experiment.