Generic and consistent confidence and credible regions

Abstract

A generic, consistent, efficient and exact method is proposed for set selection. The method is generic in that its definition and implementation uses only the likelihood function. The method is consistent in that the same criterion is used to select confidence and credible sets making the two kinds of sets consistent even though the two sets may differ since they answer different questions. The method is exact in that no approximations are used except numerical integration which can be made as exact as needed by investing computational resources. The method is efficient to the point that it makes possible confidence set determinations by numerical integration which are otherwise impractical. The method requires computational resources comparable to what is needed for a Bayesian analysis and may be more efficient than bootstrap of maximum likelihood estimates as it avoids repeated minimizations of randomly perturbed data. Central to the proposed approach are the use of (1) reference priors (e.g., Bernardo, 2005), (2) pointwise mutual information as test statistics and (3) importance sampling to efficiently evaluate series of related statistical integrals (e.g., Schafer, 2009). These central pieces are expected to be useful to address statistical questions beyond set selection.