L-cumulants, L-cumulant embeddings and algebraic statistics

Focusing on the discrete probabilistic setting we generalize the combinatorial definition of cumulants. Obtained in this way L-cumulants keep 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 in 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. Some recent papers show that this way of thinking about geometric problems can be very rewarding.