  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.
