Posterior contraction rates for support boundary recovery

TL;DR

This paper investigates the rate at which Bayesian methods can accurately recover a boundary function in a Poisson point process model, providing theoretical guarantees for different prior choices.

Contribution

It establishes a general posterior contraction rate result for boundary recovery in a non-standard Poisson process model, including specific cases for Gaussian and wavelet priors.

Findings

Derived a general posterior contraction rate with respect to the $L^1$-norm.

Applied the result to Gaussian process priors.

Applied the result to wavelet series priors.

Abstract

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 establish a general result for the posterior contraction rate with respect to the $L^1$-norm based on entropy and one-sided small probability bounds. From this, specific posterior contraction results are derived for Gaussian process priors and priors based on random wavelet series.