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This paper proposes a new methodology to perform Bayesian inference for aclass of multidimensional Cox processes in which the intensity function ispiecewise constant. Poisson processes with piecewise constant intensityfunctions are believed to be suitable to model a variety of point processphenomena and, given its simpler structure, are expected to provide moreprecise inference when compared to processes with non-parametric andcontinuously varying intensity functions. The partition of the space domain isflexibly determined by a level-set function of a latent Gaussian process.Despite the intractability of the likelihood function and the infinitedimensionality of the parameter space, inference is performed exactly, in thesense that no space discretization approximation is used and MCMC error is theonly source of inaccuracy. That is achieved by using retrospective samplingtechniques and devising a pseudo-marginal infinite-dimensional MCMC algorithmthat converges to the exact target posterior distribution. Computationalefficiency is favored by considering a nearest neighbor Gaussian process,allowing for the analysis of large datasets. An extension to considerspatiotemporal models is also proposed. The efficiency of the proposedmethodology is investigated in simulated examples and its applicability isillustrated in the analysis of some real point process datasets.

Exact Bayesian inference for level-set Cox processes with piecewise constant intensity function