Coverage of credible intervals in nonparametric monotone regression

Abstract

For nonparametric univariate regression under a monotonicity constraint on the regression function f, we study the coverage of a Bayesian credible interval for $\mathit{f}({\mathit{x}}_0)$, where ${\mathit{x}}_0$ is an interior point. Analysis of the posterior becomes a lot more tractable by considering a “projection-posterior” distribution based on a finite random series of step functions with normal basis coefficients as a prior for f. A sample f from the resulting conjugate posterior distribution is projected on the space of monotone increasing functions to obtain a monotone function ${\mathit{f}}^{\ast }$ closest to f, inducing the “projection-posterior.” We use projection-posterior samples to obtain credible intervals for $\mathit{f}({\mathit{x}}_0)$. We obtain the asymptotic coverage of the credible interval thus constructed and observe that it is free of nuisance parameters involving the true function. We observe a very interesting phenomenon that the coverage is typically higher than the nominal credibility level, the opposite of a phenomenon observed by Cox (Ann. Statist. 21 (1993) 903–923) in the Gaussian sequence model. We further show that a recalibration gives the right asymptotic coverage by starting from a lower credibility level that can be explicitly calculated.