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：本文研究了非齐次泊松点过程强度函数的非参数贝叶斯估计，重点分析了强度依赖于协变量的重要情形，基于大面积区域上单个点模式实现的观测数据。研究表明，协变量的存在使得观测窗口内远距离位置的信息得以借用，从而在增长域渐近框架下实现一致推断。特别地，本文推导了全局损失与逐点损失函数下的最优后验收缩速率。全局损失下的速率基于类似成熟贝叶斯非参数理论中的先验分布条件获得，此处结合平稳过程泛函的集中不等式以控制分析中出现的特定随机协变依赖损失函数。局部速率则通过一项专门研究推导，该研究建立在Pólya树先验理论的最新进展之上，并借助协变量诱导的随机几何构造，将其拓展至当前多元情形。