L-cumulants, L-cumulant embeddings and algebraic statis- tics

Abstract

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