L-cumulants, L-cumulant embeddings and algebraic statistics

L-cumulants, L-cumulant embeddings and algebraic statistics AS2012 Special Volume, part 1: This issue includes a second series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July 2012. Zwiernik, Piotr Creator Faculty piotr.zwiernik@upf.edu MATH / Applied Mathematics Illinois Institute of Technology Affiliated department Conditional independence models discrete random variables cumulants free cumulants Boolean cumulants tree cumulants central moments 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 alge- braic 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. Article