Fast and scalable non-parametric Bayesian inference for Poisson point processes

Artikel i övrig tidskrift, 2019

We study the problem of non-parametric Bayesian estimation of the intensity function of a Poisson point process. The observations are n independent realisations of a Poisson point process on the interval [0T]. We propose two related approaches. In both approaches we model the intensity function as piecewise constant on N bins forming a partition of the interval [0T]. In the rst approach the coe cients of the intensity function are assigned independent gamma priors, leading to a closed form posterior distribution. On the theoretical side, we prove that as n the posterior asymptotically concentrates around the true , data-generating intensity function at an optimal rate for h-Holder regular intensity functions (0 < h 1). In the second approach we employ a gamma Markov chain prior on the coefcients of the intensity function. The posterior distribution is no longer available in closed form, but inference can be performed using a straightforward version of the Gibbs sampler. Both approaches scale well with sample size, but the second is much less sensitive to the choice of N. Practical performance of our methods is rst demonstrated via synthetic data examples. We compare our second method with other existing approaches on the UKcoal mining disasters data. Furthermore, we apply it to the US mass shootings data and Donald Trumps Twitter data. Keywords and phrases: Empirical Bayes, Intensity function, gamma Markov chain prior, Gibbs sampler, Independent gamma prior, Markov Chain Monte Carlo, Metropolis-within-Gibbs, Non-homogeneous Poisson process, Non-parametric Bayesian estimation, Poisson point process, Posterior contraction rate.