摘要：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 n−1∕3 under the L1-metric and at a nearly optimal rate under the empirical Lp-metrics for 0关键词：Bayesian testing; Monotonicity; posterior contraction; projection-posterior
