Fast and scalable non-parametric Bayesian inference for Poisson point processes
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We study the problem of non-parametric Bayesian estimation of the intensity function of a Poisson point process. The observations are assumed to be n independent realisations of a Poisson point process on the interval [0,T]. 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 [0,T]. In the first approach the coefficients of the intensity function are assigned independent Gamma priors. This leads to a closed form posterior distribution, for which posterior inference is straightforward to perform in practice, without need to recourse to approximate inference methods. The method scales extremely well with the amount of data. On the theoretical side, we prove that the approach is consistent: as n→∞, the posterior distribution asymptotically concentrates around the "true", data-generating intensity function at the rate that is optimal for estimating h-Hölder regular intensity functions (0 < h≤ 1), provided the number of coefficients N of the intensity function grows at a suitable rate depending on the sample size n. In the second approach it is assumed that the prior distribution on the coefficients of the intensity function forms a Gamma Markov chain. The posterior distribution is no longer available in closed form, but inference can be performed using a straightforward version of the Gibbs sampler. We show that also this second approach scales well. Practical performance of our methods is first demonstrated via synthetic data examples. It it shown that the second approach depends in a less sensitive way on the choice of the number of bins N and outperforms the first approach in practice. Finally, we analyse three real datasets using our methodology: the UK coal mining disasters data, the US mass shootings data and Donald Trump's Twitter data.
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