Nonparametric Bayesian analysis for support boundary recovery

Given a sample of a Poisson point process with intensity $\lambda_f(x,y) = n \mathbf{1}(f(x) \leq y),$ we study recovery of the boundary function $f$ from a nonparametric Bayes perspective. Because of the irregularity of this model, the analysis is non-standard. We derive contraction rates with respect to the $L^1$-norm for several classes of priors, including Gaussian priors, priors based on (truncated) random series, compound Poisson processes, and subordinators. We also investigate the limiting shape of the posterior distribution and derive a nonparametric version of the Bernstein-von Mises theorem for a specific class of priors on a function space with increasing parameter dimension. We show that the marginal posterior of the functional $\vartheta =\int f$ does some automatic bias correction and contracts with a faster rate than the MLE. In this case, $1-\alpha$-credible sets are also asymptotic $1-\alpha$ confidence intervals. It is also shown that the frequentist coverage of credible sets is lost under model misspecification.