  We discuss Bayesian inference for parameters selected using the data. First,
we provide a critical analysis of the existing positions in the literature
regarding the correct Bayesian approach under selection. Second, we propose two
types of non-informative priors for selection models. These priors may be
employed to produce a posterior distribution in the absence of prior
information as well as to provide well-calibrated frequentist inference for the
selected parameter. We test the proposed priors empirically in several
scenarios.
