Rates and coverage for monotone densities using projection-posterior

Abstract

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 n−1/3-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.