Poisson Log-Normal (PoLoN) Process for non-parametric prediction of count data

Modeling datasets of integer counts is crucial in physics and other scientific disciplines, where measurements involve discrete, non-negative quantities. Traditional approaches such as Poisson regression often struggle to capture complex, non-linear relationships due to their fixed parametric forms. Non-parametric Gaussian Process (GP) regression, while providing a powerful framework for modeling continuous data, is less suited to count data because it assumes Gaussian-distributed outputs unconstrained by discreteness or non-negativity. We propose the Poisson Log-Normal (PoLoN) process, which combines GP flexibility with the Poisson-lognormal distribution to model integer count data. Our method assumes that the observed counts follow a Poisson distribution with an input-dependent rate parameter whose logarithm is modeled by a GP, capturing correlations through the kernel function rather than explicit parameterization. PoLoN provides a unified framework for signal extraction and predictive modeling, enabling separation of localized signal peaks from smooth backgrounds and accurate prediction of integer-valued data. Tests on synthetic and real datasets, including bike rentals and Higgs-Boson discovery data, demonstrate PoLoN's robustness in handling complex, structured data across scientific domains.