Bayesian Encompassing Specification Tests of a Parametric Model Against a Nonparametric Alternative

Encompassing tests of a model \(M_0\) are based upon the notion that \(M_0\) ought to be able to account for results derived upon alternative models. Within a Bayesian framework, posterior distributions obtained under an alternative model \(M_1\) are explicitly compared with posterior distributions obtained within \(M_0\) by means of a suitable “transition” distribution for the parameters of \(M_1\) conditionnally on those of \(M_0\) . In our chapter, we propose a generic Bayesian procedure for testing the capabilities of a parametric model \(M_0\) to encompass a wide range of inferential results derived under a nonparametric alternative \(M_1\) . By combining the use of Dirichlet prior measures on \(M_1\) with that of Monte Carlo simulation techniques, we develop exact operational and highly flexible Bayesian encompassing test procedures of \(M_0\) relative to \(M_1\) . An application to an exponential lifetime model \(M_0\) illustrates the performance of our procedure.