Shape restrictions such as monotonicity on functions often arise naturally in statistical modeling. We consider a Bayesian approach to the estimation of a monotone regression function and testing for monotonicity. We construct a prior distribution using piecewise constant functions. For estimation, a prior imposing monotonicity of the heights of these steps is sensible, but the resulting posterior is harder to analyze theoretically. We consider a “projection-posterior” approach, where a conjugate normal prior is used, but the monotonicity constraint is imposed on posterior samples by a projection map onto the space of monotone functions. We show that the resulting posterior contracts at the optimal rate under the -metric and at a nearly optimal rate under the empirical -metrics for . The projection-posterior approach is also computationally more convenient. We also construct a Bayesian test for the hypothesis of monotonicity using the posterior probability of a shrinking neighborhood of the set of monotone functions. We show that the resulting test has a universal consistency property and obtain the separation rate which ensures that the resulting power function approaches one.
Research is partially supported by NSF grant number DMS-1916419.
"Convergence rates for Bayesian estimation and testing in monotone regression." Electron. J. Statist. 15 (1) 3478 - 3503, 2021. https://doi.org/10.1214/21-EJS1861