This work studies nonparametric Bayesian estimation of the intensity function of an inhomogeneous Poisson point process in the important case where the intensity depends on covariates, based on the observation of a single realisation of the point pattern over a large area. It is shown how the presence of covariates allows to borrow information from far away locations in the observation window, enabling consistent inference in the growing domain asymptotics. In particular, optimal posterior contraction rates under both global and point-wise loss functions are derived. The rates in global loss are obtained under conditions on the prior distribution resembling those in the well established theory of Bayesian nonparametrics, here combined with concentration inequalities for functionals of stationary processes to control certain random covariate-dependent loss functions appearing in the analysis. The local rates are derived with an ad-hoc study that builds on recent advances in the theory of P\'olya tree priors, extended to the present multivariate setting with a novel construction that makes use of the random geometry induced by the covariates.

翻译：本研究探讨了在强度函数依赖于协变量的重要情形下，基于单个大区域点模式实现观察，对非齐次泊松点过程强度函数进行非参数贝叶斯估计。研究表明，协变量的存在使得能从观测窗口内相距较远的位置借用信息，从而在区域增长渐近框架下实现一致推断。具体而言，推导了全局损失函数与逐点损失函数下的最优后验收缩速率。全局损失下的速率的获得依赖于对先验分布施加的条件，这些条件类似于贝叶斯非参数成熟理论中的条件，并结合平稳过程泛函的集中不等式来控制分析中出现的特定随机协变量依赖损失函数。局部损失速率通过一项专门研究获得，该研究基于波利亚树先验理论的最新进展，并通过利用协变量诱导的随机几何构造的新方法，将其推广至当前多元背景。