  Suppose that a compound Poisson process is observed discretely in time and
assume that its jump distribution is supported on the set of natural numbers.
In this paper we propose a non-parametric Bayesian approach to estimate the
intensity of the underlying Poisson process and the distribution of the jumps.
We provide a MCMC scheme for obtaining samples from the posterior. We apply our
method on both simulated and real data examples, and compare its performance
with the frequentist plug-in estimator proposed by Buchmann and Gr\"ubel. On a
theoretical side, we study the posterior from the frequentist point of view and
prove that as the sample size $n\rightarrow\infty$, it contracts around the
`true', data-generating parameters at rate $1/\sqrt{n}$, up to a $\log n$
factor.
