  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.
