Adjusted Bayesian inference for selected parameters

TL;DR

This paper develops a Bayesian framework for inference on parameters selected after data viewing, addressing the need for adjustment in cases of non-informative priors or fixed parameters, and introduces a Bayesian FDR control method.

Contribution

It introduces a Bayesian adjustment method for post-selection inference and a Bayesian FDR control approach, extending existing methods beyond the two-group model.

Findings

Adjusted Bayesian inference is necessary for non-informative priors or fixed parameters.

The proposed Bayesian FDR method generalizes existing approaches.

Application to simulated and microarray data demonstrates effectiveness.

Abstract

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 FDR methods that are only defined in the two-group mixture model.We illustrate our results by applying them to simulated data and data froma microarray experiment.