  This paper proposes a new methodology to perform Bayesian inference for a
class of multidimensional Cox processes in which the intensity function is
piecewise constant. Poisson processes with piecewise constant intensity
functions are believed to be suitable to model a variety of point process
phenomena and, given its simpler structure, are expected to provide more
precise inference when compared to processes with non-parametric and
continuously varying intensity functions. The partition of the space domain is
flexibly determined by a level-set function of a latent Gaussian process.
Despite the intractability of the likelihood function and the infinite
dimensionality of the parameter space, inference is performed exactly, in the
sense that no space discretization approximation is used and MCMC error is the
only source of inaccuracy. That is achieved by using retrospective sampling
techniques and devising a pseudo-marginal infinite-dimensional MCMC algorithm
that converges to the exact target posterior distribution. Computational
efficiency is favored by considering a nearest neighbor Gaussian process,
allowing for the analysis of large datasets. An extension to consider
spatiotemporal models is also proposed. The efficiency of the proposed
methodology is investigated in simulated examples and its applicability is
illustrated in the analysis of some real point process datasets.
