Higher order asymptotic computation of Bayesian significance tests for precise none hypotheses in the presence of nuisance parameters

The full Bayesian significance test (FBST) was introduced by Pereira and Stern for measuring the evidence of a precise none hypothesis. The FBST requires both numerical optimization and multidimensional integration, whose computational cost may be heavy when testing a precise none hypothesis on a scalar parameter of interest in the presence of a large number of nuisance parameters. In this paper we propose a higher order approximation of the measure of evidence for the FBST, based on tail area expansions of the marginal posterior of the parameter of interest. When in particular focus is on matching priors, further results are highlighted. Numerical illustrations are discussed.This work was supported the Cariparo Foundation Excellence grant 2011/2012, and by grants from Regione Autonoma della Sardegna (CRP-59903) and Ministerio de Economia y Competitividad of Spain (ECO2012-38442, RYC-2012- 11455)