Efficient posterior sampling for Bayesian Poisson regression

TL;DR

This paper introduces efficient Metropolis-Hastings and importance sampling algorithms for Bayesian Poisson regression, leveraging Gaussian approximations and Pólya-gamma augmentation to outperform existing methods in computational speed and efficiency.

Contribution

The authors develop novel sampling algorithms that significantly improve the efficiency of Bayesian Poisson regression, using Gaussian approximations and Pólya-gamma data augmentation.

Findings

Proposed samplers are faster than traditional methods.

Metropolis-Hastings outperforms Hamiltonian Monte Carlo in all tested scenarios.

Algorithms are competitive in terms of time per independent sample.

Abstract

Poisson log-linear models are ubiquitous in many applications, and one of the most popular approaches for parametric count regression. In the Bayesian context, however, there are no sufficient specific computational tools for efficient sampling from the posterior distribution of parameters, and standard algorithms, such as random walk Metropolis-Hastings or Hamiltonian Monte Carlo algorithms, are typically used. Herein, we developed an efficient Metropolis-Hastings algorithm and importance sampler to simulate from the posterior distribution of the parameters of Poisson log-linear models under conditional Gaussian priors with superior performance with respect to the state-of-the-art alternatives. The key for both algorithms is the introduction of a proposal density based on a Gaussian approximation of the posterior distribution of parameters. Specifically, our result leverages the negative binomial approximation of the Poisson likelihood and the successful P\'olya-gamma data augmentation scheme. Via simulation, we obtained that the time per independent sample of the proposed samplers is competitive with that obtained using the successful Hamiltonian Monte Carlo sampling, with the Metropolis-Hastings showing superior performance in all scenarios considered.