Bayesian testing for embedded hypotheses with application to shape constrains

In this paper we study Bayesian answers to testing problems when the hypotheses are not well separated and propose a general approach with a special focus on shape constrains testing. We then apply our method to several testing problems including testing for positivity and monotonicity in a nonparametric regression setting. For each of this problems, we show that our approach leads to the optimal separation rate of testing, which indicates that our tests have the best power. To our knowledge, separation rates have not been studied in the Bayesian literature so far.