  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.
