Most objective Bayesian hypothesis tests result in exponential accumulation of evidence in favor of true alternative hypotheses, but only sub-linear accumulation of evidence in favor of true point null hypotheses. Thus, it is often impossible for such tests to provide strong evidence in favor of a true null hypothesis, even when moderately large sample sizes have been obtained. Because Bayesian hypothesis tests yield probability statements regarding the truth of the null hypothesis (rather than a frequentist decision to simply not reject the hypothesis), the resulting imbalance in the rates of accumulation of evidence is problematic. In this article, we review asymptotic convergence rates of standard objective Bayes factors and propose two new classes of prior densities that ameliorate the imbalance in convergence rates inherited by standard objective methods. Using members of these classes, we obtain analytic expressions for Bayes factors in linear models and derive approximations to the resulting Bayes factors in large-sample settings.
Biostatistics | Clinical Trials | Statistical Methodology | Statistical Theory
Johnson, Valen E. and Rossell, David, "Non-Local Prior Densities for Default Bayesian Hypothesis Tests" (May 2008). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 42.