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 factor of the parametric model to the nonparametric alternative. Doing so by specifying the nonparametric alternative via a Polya tree process is particularly attractive, from both theoretical and methodological perspectives. Among the benefits is a degree of computational simplicity that even allows for robustness analyses to be implemented. Default (non-subjective) versions of this analysis are developed herein, in the sense that recommended choices are provided for the (many) features of the Polya tree process that need to 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 be viewed as a conditional frequentist test, resulting in data-dependent reported error probabilities that have a real frequentist interpretation (as opposed to p-values) in even small sample situations.