Exact Bayesian inference for level-set Cox processes

This paper proposes a class of multidimensional Cox processes in which the intensity function is piecewise constant and develops a methodology to perform Bayesian inference without the need to resort to discretisation-based approximations. 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 provided more precise inference when compared to processes with non-parametric intensity functions that vary continuously across the space. The piecewise constant property is determined by a level-set function of a latent Gaussian process so that the regions in which the intensity function is constant are defined in a flexible manner. 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 discretisation approximation is used and Monte Carlo 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. 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.