L-cumulants, L-cumulant embeddings and algebraic statistics

TL;DR

This paper introduces L-cumulants, a generalization of classical cumulants, which retain key properties and enhance algebraic analysis of statistical models like Markov and hidden Markov models, especially with binary hidden states.

Contribution

It defines L-cumulants in the discrete setting, demonstrating their usefulness in algebraic statistics and providing insights into binary hidden tree models and algebraic varieties.

Findings

L-cumulants preserve semi-invariance and independence properties.

Application to Markov and hidden Markov models with binary hidden states.

Enhanced understanding of algebraic structures in statistical models.

Abstract

Focusing on the discrete probabilistic setting we generalize the combinatorial definition of cumulants to L-cumulants. This generalization keeps all the desired properties of the classical cumulants like semi-invariance and vanishing for independent blocks of random variables. These properties make L-cumulants useful for the algebraic analysis of statistical models. We illustrate this for general Markov models and hidden Markov processes in the case when the hidden process is binary. The main motivation of this work is to understand cumulant-like coordinates in algebraic statistics and to give a more insightful explanation why tree cumulants give such an elegant description of binary hidden tree models. Moreover, we argue that L-cumulants can be used in the analysis of certain classical algebraic varieties.