  We propose a general modeling framework for marked Poisson processes observed
over time or space. The modeling approach exploits the connection of the
nonhomogeneous Poisson process intensity with a density function. Nonparametric
Dirichlet process mixtures for this density, combined with nonparametric or
semiparametric modeling for the mark distribution, yield flexible prior models
for the marked Poisson process. In particular, we focus on fully nonparametric
model formulations that build the mark density and intensity function from a
joint nonparametric mixture, and provide guidelines for straightforward
application of these techniques. A key feature of such models is that they can
yield flexible inference about the conditional distribution for multivariate
marks without requiring specification of a complicated dependence scheme. We
address issues relating to choice of the Dirichlet process mixture kernels, and
develop methods for prior specification and posterior simulation for full
inference about functionals of the marked Poisson process. Moreover, we discuss
a method for model checking that can be used to assess and compare goodness of
fit of different model specifications under the proposed framework. The
methodology is illustrated with simulated and real data sets.
