May 2022 Rates and coverage for monotone densities using projection-posterior
Moumita Chakraborty, Subhashis Ghosal
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Bernoulli 28(2): 1093-1119 (May 2022). DOI: 10.3150/21-BEJ1379


We consider Bayesian inference for a monotone density on the unit interval and study the resulting asymptotic properties. We consider a “projection-posterior” approach, where we construct a prior on density functions through random histograms without imposing the monotonicity constraint, but induce a random distribution by projecting a sample from the posterior on the space of monotone functions. The approach allows us to retain posterior conjugacy, allowing explicit expressions extremely useful for studying asymptotic properties. We show that the projection-posterior contracts at the optimal n1/3-rate. We then construct a consistent test based on the posterior distribution for testing the hypothesis of monotonicity. Finally, we obtain the limiting coverage of a projection-posterior credible interval for the value of the function at an interior point. Interestingly, the limiting coverage turns out to be higher than the nominal credibility level, the opposite of the undercoverage phenomenon observed in a smoothness regime. Moreover, we show that a recalibration method using a lower credibility level gives an intended limiting coverage. We also discuss extensions of the obtained results for densities on the half-line. We conduct a simulation study to demonstrate the accuracy of the asymptotic results in finite samples.

Funding Statement

The second author’s research is partially supported by NSF Grant number DMS-1916419.


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Moumita Chakraborty. Subhashis Ghosal. "Rates and coverage for monotone densities using projection-posterior." Bernoulli 28 (2) 1093 - 1119, May 2022.


Received: 1 August 2020; Revised: 1 January 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

MathSciNet: MR4388931
zbMATH: 07526577
Digital Object Identifier: 10.3150/21-BEJ1379

Keywords: Bayesian test for monotonicity , contraction rate , coverage , credible interval , monotone density

Rights: Copyright © 2022 ISI/BS


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Vol.28 • No. 2 • May 2022
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