Adjusted Bayesian inference for selected parameters

We address the problem of providing inference for parameters selected after viewing the data. A frequentist solution to this problem is False Discovery Rate adjusted inference. We explain the role of selection in controlling the occurrence of false discoveries in Bayesian analysis, and argue that Bayesian inference may also be affected by selection – in particular Bayesian inference based on subjective priors. We introduce selection-adjusted Bayesian methodology based on the conditional posterior distribution of the parameters given selection; show how it can be used to specify selection criteria; explain how it relates to the Bayesian FDR approach; and apply it to microarray data.