Adaptive truncation and simulation-based calibration applied to Com-Poisson models
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2026-03-27
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Carvalho, Luiz Max Fagundes de
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Count data are ubiquitous across various scientific and social domains, yet the standard Poisson distribution often fails to capture the over-dispersion or under-dispersion frequently observed in real-world scenarios. The Conway-Maxwell-Poisson (COM-Poisson) distribution offers a flexible alternative by incorporating a dispersion parameter; however, its practical application is hindered by the computational complexity of its normalising constant, which involves an infinite sum. This work addresses this challenge by developing and implementing adaptive truncation strategies with verified error bounds, ensuring that the normalising constant can be computed to any desired level of precision. The research introduces a framework for computational faithfulness in Bayesian inference, utilising Simulation-Based Calibration (SBC) to validate the entire modelling pipeline. By systematically varying numerical precision, the study demonstrates that insufficient accuracy in the normalising constant does not merely introduce noise but can lead to systematic biases and “computational collapse” in the Markov Chain Monte Carlo (MCMC) samplers. The proposed methodology is implemented in R and Stan languages; the truncated methods are implemented also in InfSumPy library and the COM-Poisson approach integrated into widely used brms R package. Empirical validation across diverse datasets, ranging from ecology and linguistics to retail and finance, confirms the model’s versatility and the critical importance of numerical precision for robust statistical inference.