Nonparametric Bayesian analysis of the compound Poisson prior for support boundary recovery

TL;DR

This paper studies Bayesian methods for support boundary recovery in Poisson point process data, demonstrating near-optimal posterior contraction, bias correction, and limitations of credible sets for certain functionals.

Contribution

It introduces nonparametric Bayesian analysis with compound Poisson priors, showing adaptive contraction rates and a novel Bernstein-von Mises result for increasing dimension models.

Findings

Posterior contracts at nearly optimal rates for monotone and piecewise constant boundaries.

The marginal posterior of the integral functional contracts faster than the MLE, with credible sets serving as confidence intervals.

Credible sets lack frequentist coverage for linear functionals unless priors are specifically matched to the true function.

Abstract

Given data from a Poisson point process with intensity $(x,y) \mapsto n \mathbf{1}(f(x)\leq y),$ frequentist properties for the Bayesian reconstruction of the support boundary function $f$ are derived. We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone and piecewise constant support boundaries and adapts to H\"older smooth boundaries with smoothness index at most one. We then derive a non-standard Bernstein-von Mises result for a compound Poisson process prior and a function space with increasing parameter dimension. As an intermediate result the limiting shape of the posterior for random histogram type priors is obtained. In both settings, it is shown that the marginal posterior of the functional $\vartheta =\int f$ performs an 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. As a negative result, it is shown that the frequentist coverage of credible sets is lost for linear functions indicating that credible sets only have frequentist coverage for priors that are specifically constructed to match properties of the underlying true function.