I discuss the impact of three principles on the problem of choosing a good goodness-of-fit test. First: when testing statistical hypotheses alternatives of interest are neither indetectably nor grossly different from the null hypothesis. Second, good tests are designed to be sensitive to alternatives likely to arise in practice. Third, the purpose of limit theorems is to provide good approximate probability calculations of interest to statisticians.
I will use Bayesian priors on the alternative hypothesis to construct tests which maximize the expected power for a prior which depends on the sample size. Priors will be presented for which the optimal procedures are (approximately) such goodness-of-fit tests as the Cramér-von Mises or the Anderson-Darling test.
My colleagues and I have formalized specific scientific questions in terms of point process intensity functions, and have used Bayesian methods to fit the point process models to neuronal data (though we sometimes prefer simple smoothers and the Bootstrap). In my talk I will very briefly outline some of the substantive problems we are examining and the progress being made. I will also give some details on BARS (Bayesian Adaptive Regression Splines), an approach to generalized nonparametric regression which we have found quite useful.