Rate-optimal Bayesian intensity smoothing for inhomogeneous Poisson processes

Abstract

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 analysing count data coming from a call centre 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.

Posterior Concentration Rates for Counting Processes with Aalen Multiplicative Intensities

“…The proof of Theorem 2.1 is reported in Section 4. To the best of our knowledge, the only other paper dealing with posterior concentration rates in related models is that of Belitser et al (2013), where inhomogeneous Poisson processes are considered. Theorem 2.1 differs in two aspects from their Theorem 1.…”

Estimation of the intensity function of an inhomogeneous Poisson process with a change‐point

“…; Samo & Roberts, ), kernel mixtures priors (Kottas & Sansó, ) and spline priors (Palacios & Minin, ; Belitser et al . ) have been developed. Posterior convergence and contraction rates of intensity function under general and spline prior settings have been investigated by Belitser et al .…”

“…Posterior convergence and contraction rates of intensity function under general and spline prior settings have been investigated by Belitser et al . (), while convergence rates under Gaussian process priors have been studied by Kirichenko & van Zanten ().…”

“…Previous work has considered the case where the true intensity is a continuous function (Belitser et al . ) or Hölder continuous function (Kirichenko & van Zanten, ). For example, failures of repairable systems are often described by means of non‐homogeneous Poisson processes which implicitly assume that the reliability of a system evolves continuously over time.…”

“…We derive posterior convergence rates of the intensity function and the change‐point under the general prior setting with similar assumptions to Belitser et al . (), and also for the special case of a Gaussian process prior. As argued by Diaconis & Freedman (), these non‐parametric methods are of little value unless they possess reasonable consistency properties.…”