A Gaussian Cox process is a popular model for point process data, in which the intensity function is a transformation of a Gaussian process. Posterior inference of this intensity function involves an intractable integral (i.e., the cumulative intensity function) in the likelihood resulting in doubly intractable posterior distribution. Here, we propose a nonparametric Bayesian approach for estimating the intensity function of an inhomogeneous Poisson process without reliance on large data augmentation or approximations of the likelihood function. We propose to jointly model the intensity and the cumulative intensity function as a transformed Gaussian process, allowing us to directly bypass the need of approximating the cumulative intensity function in the likelihood. We propose an exact MCMC sampler for posterior inference and evaluate its performance on simulated data. We demonstrate the utility of our method in three real-world scenarios including temporal and spatial event data, as well as aggregated time count data collected at multiple resolutions. Finally, we discuss extensions of our proposed method to other point processes.

翻译：高斯Cox过程是点过程数据的常用模型，其强度函数为高斯过程的变换形式。该强度函数的后验推断涉及似然函数中难以处理的积分（即累积强度函数），导致后验分布具有双重难解性。本文提出一种非参数贝叶斯方法，用于估计非齐次泊松过程的强度函数，该方法无需依赖大规模数据增广或似然函数近似。我们建议将强度函数与累积强度函数作为变换后的高斯过程进行联合建模，从而直接规避似然函数中累积强度函数近似的需求。我们提出一种精确的MCMC采样器用于后验推断，并在模拟数据上评估其性能。通过时态与空间事件数据，以及多分辨率采集的聚合时间计数数据等三种实际场景，验证了本方法的实用性。最后，我们探讨了所提方法向其他点过程的扩展。