[Paper Review] Fast and scalable non-parametric Bayesian inference for Poisson point processes

Shota Gugushvili, Frank van der Meulen+2 more|arXiv (Cornell University)|Apr 10, 2018

Statistical Methods and Bayesian Inference4 citations

TL;DR

This paper proposes two fast, scalable non-parametric Bayesian methods for estimating the intensity function of a non-homogeneous Poisson point process on [0,T]. The first uses independent gamma priors for piecewise-constant intensity on N bins, yielding a closed-form posterior; the second employs a gamma Markov chain prior enabling MCMC inference via Gibbs sampling. The key contribution is theoretical posterior contraction at the optimal rate for h-Hölder continuous intensities, with the second method showing superior robustness to bin count choice.

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 $n ightarrow\infty,$ the posterior asymptotically concentrates around the "true", data-generating intensity function at an optimal rate for $h$-Hölder 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.

Motivation & Objective

• To develop fast and scalable non-parametric Bayesian methods for estimating the intensity function of a Poisson point process.
• To ensure theoretical optimality in posterior concentration rates for h-Hölder continuous intensity functions.
• To improve robustness to the choice of the number of bins N compared to existing methods.
• To enable practical inference via efficient MCMC algorithms, particularly for large datasets.

Proposed method

• Model the intensity function as piecewise constant over N bins partitioning [0,T], with coefficients assigned independent gamma priors for closed-form posterior computation.
• Use a gamma Markov chain prior on intensity coefficients to induce smoothness and enable MCMC sampling via a straightforward Gibbs sampler.
• Derive a reversible jump MCMC algorithm to explore models with varying numbers of bins N, using a prior on N and local proposals for model moves.
• Implement a Metropolis-within-Gibbs sampler for the gamma Markov chain model, with priors on the smoothing parameter α and bin coefficients.
• Use marginal likelihood approximation and trace plots to assess model selection and convergence in reversible jump MCMC.
• Apply both methods to synthetic data, UK coal mining disasters, US mass shootings, and Donald Trump’s Twitter data to demonstrate practical performance.

Experimental results

Research questions

• RQ1 Can we achieve fast and scalable non-parametric Bayesian inference for Poisson point process intensity functions?
• RQ2 Does the use of independent gamma priors on piecewise-constant intensity coefficients yield a closed-form posterior and optimal posterior contraction?
• RQ3 Can a gamma Markov chain prior improve robustness to the choice of the number of bins N compared to independent gamma priors?
• RQ4 How do the proposed methods perform on real-world datasets with complex intensity patterns, such as discontinuities or oscillations?
• RQ5 What is the behavior of the marginal likelihood and posterior model index distribution when using reversible jump MCMC with different priors on N?

Key findings

• The independent gamma prior approach yields a closed-form posterior distribution, enabling fast computation and theoretical posterior contraction at the optimal rate for h-Hölder continuous intensity functions.
• The gamma Markov chain prior approach does not yield a closed-form posterior but allows efficient inference via Gibbs sampling and is significantly less sensitive to the choice of N.
• The posterior contraction rate for both methods achieves the optimal rate for h-Hölder continuous intensity functions (0 < h ≤ 1), confirming theoretical optimality.
• In synthetic examples, the gamma Markov chain prior produces visually smoother posterior realizations and better captures complex intensity patterns, including discontinuities.
• On the UK coal mining disasters data, the second method outperforms existing approaches, particularly in boundary and high-variability regions.
• Numerical experiments show that the marginal log-likelihood has multiple local maxima when using independent gamma priors with reversible jump MCMC, indicating challenges in model space exploration.