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

  • Piotr Zwiernik Author University: University of California, Berkeley Author Department: Department of Statistics
Keywords: Conditional independence models, discrete random variables, cumulants, free cumulants, Boolean cumulants, tree cumulants, central moments

Abstract

Focusing on the discrete probabilistic setting we generalize the combinatorial denition
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.

References

Published Year: 2012
Volume: 3
Number: 1
Page Numbers: 11-43
Published
2012-04-30