An important task in the statistical analysis of inhomogeneous point processes is to investigate the influence of a set of covariates on the point-generating mechanism. In this article, we consider the nonparametric Bayesian approach to this problem, assuming that $n$ independent and identically distributed realizations of the point pattern and the covariate random field are available. In many applications, different covariates are often vastly diverse in physical nature, resulting in anisotropic intensity functions whose variations along distinct directions occur at different smoothness levels. To model this scenario, we employ hierarchical prior distributions based on multi-bandwidth Gaussian processes. We prove that the resulting posterior distributions concentrate around the ground truth at optimal rate as $n\to\infty$, and achieve automatic adaptation to the anisotropic smoothness. Posterior inference is concretely implemented via a Metropolis-within-Gibbs Markov chain Monte Carlo algorithm that incorporates a dimension-robust sampling scheme to handle the functional component of the proposed nonparametric Bayesian model. Our theoretical results are supported by extensive numerical simulation studies. Further, we present an application to the analysis of a Canadian wildfire dataset.

翻译：非均匀点过程统计分析的一项重要任务是研究一组协变量对点生成机制的影响。本文采用非参数贝叶斯方法处理该问题，假设可获得点模式与协变量随机场的$n$个独立同分布实现。在许多应用中，不同协变量的物理性质往往差异显著，导致强度函数呈现各向异性，即沿不同方向的变异发生在相异的平滑度水平上。为建模此场景，我们采用基于多带宽高斯过程的层次先验分布。我们证明当$n\to\infty$时，所得后验分布以最优速率集中于真实参数，并能自动适应各向异性平滑度。后验推断通过结合维度稳健抽样方案的Metropolis-within-Gibbs马尔可夫链蒙特卡洛算法具体实现，以处理所提非参数贝叶斯模型中的函数成分。理论结果得到大量数值模拟研究的支持。此外，我们展示了该方法在加拿大野火数据集分析中的应用。