2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/216479In this paper, we consider formal series associated with events, profiles derived from events, and statistical models that make predictions about events. We prove theorems about realizations for these formal series using the language and tools of Hopf algebras.Machine LearningRings and AlgebrasStatistics Theory62A01; 16W30text