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.

翻译：本文建议了一组多维的 Cox 进程, 强度函数在其中是小费常数, 并开发了一种方法, 进行贝叶斯的推论而不必使用离子化近似值。 Poisson 进程, 具有小费常数强度函数, 被认为适合于模拟各种点进程现象, 由于其结构更简单, 在与空间之间不断变化的非对称强度函数进程相比时, 预计将提供更精确的推论。 毛片常数常数由潜伏高斯进程的一个水平定值函数决定, 以便以灵活的方式定义强度函数常数的区域。 尽管参数空间的可能函数不易易感性和无限的多维性, 但精确地进行了推论, 因为不使用空间离异近值, Monte Carlo 错误是造成不准确性的唯一来源。 这是通过使用追溯性取样技术和设计一个与精确的目标远端分布相匹配的伪边远线的 MMC 算法而实现的。 在模拟中, 模拟了某些数据分析方法的效率, 也进行了模拟。