Abstract: Selective inference using multiple confidence intervals is an emerging area of statistical research whose importance is being realized very recently. We consider making such inference in the context of analyzing data with sparse signals in a Bayesian framework. Although the traditional posterior credible intervals are immune to selection, they can have low power in detecting the true signals because of covering no-signal too often if the sparse nature of the data is not properly taken into account. We demonstrate this phenomenon using a canonical Bayes model with the parameters of interest following a zero-inflated mixture prior. We propose a new method of constructing multiple intervals for any given selection rule taking a Bayesian decision theoretic approach under such a model. It involves the local fdr, the posterior probability of a parameter being null which is commonly used in multiple testing. It controls an overall measure of error rate, the Bayes or posterior false coverage rate, at a desired level among the selected intervals. We apply this method to the regression problem and demonstrate via simulations as well as data analyses that it is much more powerful in terms of enclosing zero less frequently than the traditional and some alternative methods.

Key words and phrases: FCR, multiple intervals, selection.