We study the problem of estimating the intensity function of a covariate-driven point process based on observations of the points and covariates over a large window. We consider the nonparametric Bayesian approach, and show that a wide class of Gaussian priors, combined with flexible link functions, achieves minimax-optimal posterior contraction rates in the increasing domain asymptotics and under the assumption that the covariates be ergodic. We also employ Besov-Laplace priors, which are popular in imaging and inverse problems due to their edge-preserving and sparsity-promoting properties. We prove that these yield optimal estimation of spatially inhomogeneous intensities belonging to Besov spaces with low integrability index. These results are based on a general concentration theorem that extends recent findings from the literature. To corroborate the theory, we provide extensive numerical simulations, implementing the considered procedures via suitable posterior sampling schemes. Further, we present two real data analyses motivated by applications in forestry and the environmental sciences.

翻译：本文研究基于大窗口内点和协变量观测的协变量驱动点过程强度函数估计问题。我们采用非参数贝叶斯方法，证明在递增域渐近性框架下且协变量满足遍历性假设时，结合灵活链接函数的一类广泛高斯先验可实现极小化最优后验收缩率。同时，我们引入因边缘保持和稀疏促进特性而在成像与反问题领域广泛应用的贝索夫-拉普拉斯先验，证明其能对属于低可积性指数贝索夫空间的空间非齐次强度进行最优估计。这些结果基于推广了文献近期发现的通用集中定理。为验证理论，我们通过适当的后验抽样方案实现所提方法，开展了大量数值模拟。此外，我们展示了两个源自林业与环境科学应用的真实数据分析案例。