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：我们研究基于大观测窗口内点和协变量观测值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，证明在递增域渐近框架下，当协变量满足遍历性假设时，结合灵活连接函数的一大类高斯先验能够达到极小化最优后验收缩速率。同时采用在图像处理和反问题领域因边缘保持与稀疏性促进特性而广受欢迎的贝索-拉普拉斯先验，证明其对属于低可积性指数贝索空间的空间非齐次强度具有最优估计性能。这些结果基于扩展文献最新发现的一般性集中定理。为验证理论，我们通过合适的后验抽样方案实施所考虑方法，进行了广泛的数值模拟。此外，我们展示了两个源于林业与环境科学应用场景的真实数据分析案例。