Convergence rates for Bayesian estimation and testing in monotone regression

Moumita Chakraborty, Subhashis Ghosal

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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 < p ≤ 2. 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.

Original languageEnglish (US)
Pages (from-to)3478-3503
Number of pages26
JournalElectronic Journal of Statistics
Issue number1
StatePublished - 2021
Externally publishedYes


  • Bayesian testing
  • Monotonicity
  • Posterior contraction
  • Projection-posterior

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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