Rate-optimal Bayesian intensity smoothing for inhomogeneous Poisson processes

We apply nonparametric Bayesian methods to study the problem of estimating the intensity function of an inhomogeneous Poisson process. To motivate our results we start by analyzing count data coming from a call center which we model as a Poisson process. This analysis is carried out using a certain spline prior. This prior is based on B-spline expansions with free knots, adapted from well-established methods used in regression, for instance. This particular prior is computationally feasible. Theoretically, we derive a new general theorem on contraction rates for posteriors in the setting of intensity function estimation which can be applied not just to this spline prior but also to a large number of other commonly used priors. Practical choices that have to be made in the construction of our concrete spline prior, such as choosing the priors on the number and the locations of the spline knots, are based on these theoretical findings. The results assert that when properly constructed, our approach yields a rate-optimal procedure that automatically adapts to the regularity of the unknown intensity function.