Nonparametric Goodness of Fit via Cross-Validation Bayes Factors

A nonparametric Bayes procedure is proposed for testing the fit of a parametric model for a distribution. Alternatives to the parametric model are kernel density estimates. Data splitting makes it possible to use kernel estimates for this purpose in a Bayesian setting. A kernel estimate indexed by bandwidth is computed from one part of the data, a training set, and then used as a model for the rest of the data, a validation set. A Bayes factor is calculated from the validation set by comparing the marginal for the kernel model with the marginal for the parametric model of interest. A simulation study is used to investigate how large the training set should be, and examples involving astronomy and wind data are provided. A proof of Bayes consistency of the proposed test is also provided.