A note on auxiliary mixture sampling for Bayesian Poisson models

Bayesian hierarchical Poisson models are an essential tool for analyzing count data. However, designing efficient algorithms to sample from the posterior distribution of the target parameters remains a challenging task. Auxiliary mixture sampling algorithms have been proposed to this aim. They involve two steps of data augmentation: the first leverages the theory of Poisson processes, and the second approximates the residual distribution of the resulting model through a mixture of Gaussian distributions. In this way, an approximate Gibbs sampler can be implemented. This strategy is particularly beneficial for latent Gaussian models, as it allows one to exploit the sparsity of the precision matrix associated with the random effects and to efficiently incorporate linear constraints. In this paper, we focus on the accuracy of the approximation step, highlighting scenarios where the mixture fails to represent accurately the true underlying distribution, leading to a lack of convergence in the algorithm. We outline key features to monitor, in order to assess if the approximation performs as intended. Building on this, we propose a robust version of the auxiliary mixture sampling algorithm. Our approach includes mechanisms for detecting approximation failures and introduces an enhanced approximation of the right tail of the auxiliary variable distribution, supplemented by a Metropolis-Hastings correction step when needed. Finally, we evaluate the proposed algorithm together with the original mixture sampling algorithms on both simulated and real datasets.