Adaptive Bayes test for monotonicity

We propose a Bayesian nonparametric approach to test for monotonicity in a regression setting. In that context, the usual Bayes factor approach gives poor results in practice. We thus study an alternative approach that is both efficient and straightforward to implement, which is a great improvement compared to the existing frequentists procedures. Furthermore we study its asymptotic properties and prove that our procedure attains the adaptive minimax separation rate for a wide variety H\"older smooth alternatives.