Bayes factors for Fieller's problem

This paper considers point null hypothesis testing when the sampling distribution belongs to a particular class, defined in Gleser & Hwang (1987). We discuss the drawbacks of frequentist and likelihood solutions and we show how proper Bayesian analysis encounters relatively similar difficulties. We explore the performance of several noninformative Bayesian approaches to testing, namely asymptotic approximations of Bayes factors and default Bayes factors. We argue that in a default Bayesian analysis of Fieller's problem the choice of the 'correct' prior distribution is crucial. Although standard and default Bayes factors based on Jeffreys' priors show, to a lesser extent, pathologies similar to those arising in a classical framework, default Bayes factors based on reference priors seem to correct the bias and provide sensible results in term of robustness and consistency.