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

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  1. Shota GugushviliMarch 29th, 2020 at 01:01 pm

    Our response to Paulo Serra's feedback.

  2. Shota GugushviliMarch 29th, 2020 at 01:01 pm

    Comments by Paulo Serra.

  3. Shota GugushviliMarch 29th, 2020 at 01:00 pm

    Our response to Adeline Samson's feedback.

  4. Shota GugushviliMarch 29th, 2020 at 01:00 pm

    Comments by Adeline Samson.

  5. Shota GugushviliMarch 29th, 2020 at 12:59 pm

    Our response to Ryan Martin's feedback.

  6. Shota GugushviliMarch 29th, 2020 at 12:58 pm

    After the status of the paper was converted to "Published", its public reviews and our responses to them got automatically removed. As they contain valuable discussions, I'll reupload the relevant pdf files.

    For a start, here are comments by Ryan Martin.

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Abstract

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 $[0,T]$. 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, leading to a closed form posterior distribution. On the theoretical side, we prove that as $n\rightarrow\infty,$ the posterior asymptotically concentrates around the ``true", data-generating intensity function at an optimal rate for $h$-H\"older regular intensity functions ($0 < h\leq 1$).

In the second approach we employ a gamma Markov chain prior on the coefficients 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 first demonstrated via synthetic data examples. We compare our second method with other existing approaches on the UK coal mining disasters data. Furthermore, we apply it to the US mass shootings data and Donald Trump's Twitter data.

Versions

➤  Version 1 (2020-02-27)

Citation

Shota Gugushvili, Frank van der Meulen, Moritz Schauer and Peter Spreij (2020). Fast and scalable non-parametric Bayesian inference for Poisson point processes. Researchers.One, https://researchers.one/articles/fast-and-scalable-non-parametric-bayesian-inference-for-poisson-point-processes/5f52699c36a3e45f17ae7dd6/v1.

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