Bayesian Conway-Maxwell-Poisson model with spike-and slab priors for dispersed count data with application to football scores

2026-07-20

AI Summary

This study addresses the limitations of traditional Poisson models in handling under- or over-dispersion commonly observed in football match count data. The authors propose a Bayesian Conway–Maxwell–Poisson (CMP) model that, for the first time, integrates a spike-and-slab prior with a CMP likelihood to probabilistically identify and threshold team-specific dispersion parameters relative to equidispersion as a baseline. To facilitate posterior inference under the doubly intractable likelihood, a tailored Metropolis-within-Gibbs sampling algorithm is developed. Empirical evaluation on English Premier League data demonstrates that the proposed approach substantially outperforms standard Poisson models, achieving markedly improved goodness-of-fit and predictive performance while effectively capturing heterogeneity in dispersion across teams.

Abstract

Statistical modeling for goals scored in football is typically achieved using the Poisson distribution and its variants. Here we propose a Bayesian framework for modeling under- and over-dispersion in count data by combining the Conway-Maxwell-Poisson (CMP) likelihood with a spikeand-slab (SAS) prior on unit-specific dispersion parameters. The proposed methodology generalizes Poisson-based count data models by treating equidispersion as an explicit baseline, and offering probabilistic quantification of departures from this regime, while simultaneously estimating their magnitude. Posterior inference is performed through a tailored Metropolis-within-Gibbs sampler that handles the doubly-intractable likelihood and provides efficient posterior exploration. The new method is examined using simulated data to confirm its ability to capture non-equidispersion, and applied to English Premier League (EPL) data. Dispersion is modeled at the team level and linked to goal-scoring behavior, and allows for thresholding mechanisms to distinguish teams based on their posterior probability of non-equidispersion. The results reveal heterogeneities in team-specific dispersion in the EPL, and demonstrate improvements in both model fit and predictive performance with respect to the standard Poisson model.

Problem

• over-dispersion
• under-dispersion
• count data
• football scores
• dispersion heterogeneity

Innovation

• Conway-Maxwell-Poisson
• spike-and-slab prior
• Bayesian inference
• over-dispersion
• Metropolis-within-Gibbs