Bayesian Gaussian Process Models for Flexible Conway-Maxwell Poisson Regression
DOI:
https://doi.org/10.31185/bsj.Vol22.Iss43.1666Keywords:
Keywords: Conway-Maxwell-Poisson (COM-Poisson), Gaussian Process Regression, Bayesian Non-parametric, Count Data, Over-dispersion, Under-dispersion, Hierarchical Bayesian Model, Markov Chain Monte Carlo (MCMC).Abstract
In quantitative science, regression models for count data are essential. Also, strict distributional assumptions limit standard methods. While solutions such as Negative Binomial regression address over-dispersion, they do not account for under-dispersion. Which is another common violation of the Poisson model's equidispersion assumption. A strong generalization that can model both phenomena is provided by Conway-Maxwell-Poisson (COM-Poisson) distribution. Another significant drawback is that it is implemented within Generalized Linear Model (GLM) framework. Also more complex, non-linear dependencies cannot be captured because it assumes log-linear relationships between covariates. Where including the rate (λ) also dispersion (v) parameters. In order to close this gap, a novel fully Bayesian non-parametric framework is introduced in this paper. Where the Gaussian Process COM-Poisson (GP-COM-Poisson) regression model.
The link functions of the rate and dispersion parameters are subject to independent Gaussian Process priors. Where letting them fluctuate as non-linear, smooth functions of covariates where model can simultaneously capture intricate functional relationships straight from the data. Also account for arbitrary dispersion thanks to this structure, Hamiltonian Monte Carlo (HMC) is used for inference, effective for GP models'high-dimensional posteriors. When compared to parametric and semi-parametric alternatives. It exhibits better performance in both function recovery (MSE) and predictive fit (WAIC) where usefulness of the model is further demonstrated by a real-data application to the ecological dataset of the Galápagos Islands. Where revealing a new covariate-driven heterogeneity in dispersion in addition to a non-linear impact of elevation on species richness. A strong and adaptable addition to the statistical toolbox is the GP-COM-Poisson model. Where more thorough and trustworthy explanations of intricate count data phenomena are provided.
References
References:
• Florez, M., Guindani, M., & Vannucci, M. (2025). Bayesian bivariate Conway–Maxwell–Poisson regression model for correlated count data in sports. Journal of Quantitative Analysis in Sports, 21(1), 51-71.
• Amin, M., Ashraf, B., & Siddiqa, S. M. (2025). Liu estimation method in the zero-inflated Conway Maxwell Poisson regression model. Journal of Statistical Theory and Applications, 24(1), 71-90
• Chicheli, J., Jere, S., Hamweene, O., & Moyo, E. (2025). Modelling Malaria Cases using Count Analysis Approach: COM-Poisson INGARCH and NB-INGARCH.
• Louzada, F., Ribeiro Diniz, C. A., Dawodu, O. A., & Egbon, O. A. (2025). Bayesian spatio-temporal modelling: a systematic review. Research in Statistics, 3(1), 2568891.
• Traison, T., & Vaidyanathan, V. S. (2025). Goodness-of-Fit Tests for COM-Poisson Distribution Using Stein's Characterization. Austrian Journal of Statistics, 54(2), 85-100.
• Sham, N. M., & Khoo, W. C. (2025). Conway-Maxwell-Poisson model fitting to HFMD data in Malaysia. Data Analytics and Applied Mathematics (DAAM), 1-8.
• Hoffmann, T., & Onnela, J. P. (2025). gptools: Scalable Gaussian Process Inference with Stan. Journal of Statistical Software, 112, 1-31.
• Zhang, Z., Brown, P., & Stafford, J. (2025). Efficient modeling of quasi-periodic data with seasonal Gaussian process. Statistics and Computing, 35(2), 32.
• Pan, Q., Abdulah, S., Genton, M. G., & Sun, Y. (2025). Block Vecchia Approximation for Scalable and Efficient Gaussian Process Computations. Technometrics, 1-13.
• Agyemang, E. F. (2025). A Gaussian process regression and wavelet transform time series approaches to modeling influenza A. Computers in Biology and Medicine, 184, 109367.
• Jin, B., & Xu, X. (2025). Predicting residential real estate values in china: An empirical analysis of quzhou utilizing a novel hybrid computational framework integrating gaussian process regression with bayesian hyperparameter optimization. Economics Open, 2550007.
• Greengard, P., Rachh, M., & Barnett, A. H. (2025). Equispaced Fourier representations for efficient Gaussian process regression from a billion data points. SIAM/ASA Journal on Uncertainty Quantification, 13(1), 63-89.
• Manring, I., Wang, H., Mohler, G., & Miscouridou, X. (2025). BSTPP: a python package for Bayesian spatiotemporal point processes. Journal of Applied Statistics, 1-20.
• Hamura, Y., Irie, K., & Sugasawa, S. (2025). Robust Bayesian modeling of counts with zero inflation and outliers: Theoretical robustness and efficient computation. Journal of the American Statistical Association, 1-13.
• Zhai, G., Chen, S., Yang, Z., Yang, H., Alruwaili, A., & Bobylev, N. How Do Weather and Built Environment Nonlinearly Influence Metro Ridership? A Negative Binomial Regression Enhanced by Bayesian Additive Regression Trees. A Negative Binomial Regression Enhanced by Bayesian Additive Regression Trees.
• Hussein, S. M. J. A., & Algamal, Z. Y. (2025). Generalized Ridge Estimator for Conway-Maxwell Poisson Regression Model. Iraqi Statisticians journal, 2(special issue for ICSA2025), 247-256.
• Kang, B., Hughes, J., & Haran, M. (2025). Fast Bayesian Inference for Spatial Mean-Parameterized Conway–Maxwell–Poisson Models. Journal of Computational and Graphical Statistics, 34(2), 697-706.
• Lederman, J., & Schein, A. (2025). Modeling Latent Underdispersion with Discrete Order Statistics. arXiv preprint arXiv:2507.09032.
• Nadifar, M., Bekker, A., Arashi, M., & Ramoelo, A. (2025). Bayesian Semi-Parametric Spatial Dispersed Count Model for Precipitation Analysis. arXiv preprint arXiv:2503.19117.
• Amri, A., Elvira, V., & Wilson, A. L. (2025). Particle Hamiltonian Monte Carlo. arXiv preprint arXiv:2504.09875.
• Steel, M. F., & Zens, G. (2025). Model Uncertainty in Latent Gaussian Models with Univariate Link Function. Bayesian Analysis, 1(1), 1-26.
• Corenflos, A., & Särkkä, S. (2025). Auxiliary MCMC samplers for parallelisable inference in high-dimensional latent dynamical systems. Electronic Journal of Statistics, 19(1), 1370-1424.
• Peruzzi, M., Banerjee, S., Dunson, D. B., & Finley, A. O. (2025). Gridding and Parameter Expansion for Scalable Latent Gaussian Models of Spatial Multivariate Data. Bayesian Analysis, 1(1), 1-27.
• Umam, S., Razzak, R. B., Munni, M. Y., & Rahman, A. (2025). Exploring the non-linear association of daily cigarette consumption behavior and food security-An application of CMP GAM regression. PLoS One, 20(7), e0328109.
• Sellers, K. F., & Shmueli, G. (2010). A flexible regression model for count data. Annals of Applied Statistics, 4(2), 943–961. https://doi.org/10.1214/09-AOAS323
• Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian processes for machine learning. The MIT Press.
• Sellers, K. F., & Shmueli, G. (2010). The Conway-Maxwell-Poisson distribution. arXiv:1011.2077.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 باسل فريح محمد الحرداني التخصص إحصاء رياضي
This work is licensed under a Creative Commons Attribution 4.0 International License.
