Abstract

This article 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 nonparametric 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.

摘要

本文提出了一种新的方法来对一类强度函数是分段常数的多维 Cox 过程执行贝叶斯推理。具有分段常数强度函数的泊松过程被认为适合模拟各种点过程现象，并且由于其结构更简单，与具有非参数和连续变化的强度函数的过程相比，有望提供更精确的推理。空间域的划分由潜在高斯过程的水平集函数灵活确定。尽管似然函数的难处理性和参数空间的无限维数，在没有使用空间离散化近似并且 MCMC 误差是不准确的唯一来源的意义上，推理是准确执行的。这是通过使用回顾性采样技术并设计一种收敛到精确目标后验分布的伪边缘无限维 MCMC 算法来实现的。通过考虑最近邻高斯过程来提高计算效率，从而允许分析大型数据集。还提出了考虑时空模型的扩展。所提出方法的效率在模拟示例中进行了研究，其适用性在一些实际点过程数据集的分析中得到了说明。还提出了考虑时空模型的扩展。所提出方法的效率在模拟示例中进行了研究，其适用性在一些实际点过程数据集的分析中得到了说明。还提出了考虑时空模型的扩展。所提出方法的效率在模拟示例中进行了研究，其适用性在一些实际点过程数据集的分析中得到了说明。