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.