  Given a sample from a discretely observed compound Poisson process, we
consider non-parametric estimation of the density $f_0$ of its jump sizes, as
well as of its intensity $\lambda_0.$ We take a Bayesian approach to the
problem and specify the prior on $f_0$ as the Dirichlet location mixture of
normal densities. An independent prior for $\lambda_0$ is assumed to be
compactly supported and to possess a positive density with respect to the
Lebesgue measure. We show that under suitable assumptions the posterior
contracts around the pair $(\lambda_0,f_0)$ at essentially (up to a logarithmic
factor) the $\sqrt{n\Delta}$-rate, where $n$ is the number of observations and
$\Delta$ is the mesh size at which the process is sampled. The emphasis is on
high frequency data, $\Delta\to 0$, but the obtained results are also valid for
fixed $\Delta$. In either case we assume that $n\Delta\rightarrow\infty$. Our
main result implies existence of Bayesian point estimates converging (in the
frequentist sense, in probability) to $(\lambda_0,f_0)$ at the same rate.
  We also discuss a practical implementation of our approach. The computational
problem is dealt with by inclusion of auxiliary variables and we develop a
Markov Chain Monte Carlo algorithm that samples from the joint distribution of
the unknown parameters in the mixture density and the introduced auxiliary
variables. Numerical examples illustrate the feasibility of this approach.
