Projected Wasserstein-Fisher-Rao Flows for Hawkes Background-Intensity Estimation
Abstract
We consider the problem of estimating the background intensity of a Hawkes process when the triggering mechanism is known, treating the background as an unknown finite positive measure with a regular density. We propose a likelihood-based variational approach based on the Wasserstein-Fisher-Rao geometry. In this setting, the relevant first variations are naturally distribution-valued and therefore fall outside the classical gradient-flow formulation. We develop a distributional framework for the resulting dynamics and give them a finite-dimensional meaning through projections onto statistical manifolds. We identify the true background as the unique zero of a limiting generalized gradient and, under suitable stability, compactness, integrability and approximation conditions, establish a conditional consistency theorem for measurable estimators satisfying the empirical projected equations on events whose probabilities tend to one. Finally, we specialize the construction to Gaussian-mixture manifolds and obtain an explicit finite-dimensional scheme.
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BibTeX
@misc{banales2026projected,
title = {{Projected Wasserstein-Fisher-Rao Flows for Hawkes Background-Intensity Estimation}},
author = {Isaias Banales and Arturo Jaramillo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19588},
eprint = {2610.19588},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19588},
}
RIS
TY - UNPB
TI - Projected Wasserstein-Fisher-Rao Flows for Hawkes Background-Intensity Estimation
AU - Isaias Banales
AU - Arturo Jaramillo
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19588
UR - https://arxiv.org/abs/2610.19588
AB - We consider the problem of estimating the background intensity of a Hawkes process when the triggering mechanism is known, treating the background as an unknown finite positive measure with a regular density. We propose a likelihood-based variational approach based on the Wasserstein-Fisher-Rao geometry. In this setting, the relevant first variations are naturally distribution-valued and therefore fall outside the classical gradient-flow formulation. We develop a distributional framework for the resulting dynamics and give them a finite-dimensional meaning through projections onto statistical manifolds. We identify the true background as the unique zero of a limiting generalized gradient and, under suitable stability, compactness, integrability and approximation conditions, establish a conditional consistency theorem for measurable estimators satisfying the empirical projected equations on events whose probabilities tend to one. Finally, we specialize the construction to Gaussian-mixture manifolds and obtain an explicit finite-dimensional scheme.
ER -