Bayesian and conditional frequentist testing of a parametric model versus nonparametric alternatives

Testing the fit of data to a parametric model can be done by embedding the parametric model in a nonparametric alternative and computing the Bayes fac tor of the parametric model to the nonparametric alternative. Doing so by s pecifying the nonparametric alternative via a Polya tree process is particu larly attractive, from both theoretical and methodological perspectives. Am ong the benefits is a degree of computational simplicity that even allows f or robustness analyses to be implemented Default (nonsubjective) versions o f this analysis are developed herein, in the sense that recommended choices are provided for the (many) features of the Polya tree process that need t o be specified. Considerable discussion of these features is also provided to assist those who might be interested in subjective choices. A variety of examples involving location-scale models are studied. Finally, it is shown that the resulting procedure can we viewed as a conditional frequentist te st, resulting in data-dependent reported error probabilities that have a re al frequentist interpretation (as opposed to p values) in even small sample situations.