Nonlinear Algebra from Moments and Cumulants

Moments and cumulants are quantities that measure the shape of statistical distributions and have recently gained interest from an algebraic and combinatorial point of view. In this talk we focus on two problems that are associated with them. First we try to find the moment ideal, that is the ideal parametrized by the moments of a distribution. Further, we explain how mixtures of distributions give rise to secants of the varieties in question, while local mixing corresponds to tangents. The second problem is about parameter recovery from moments or cumulants. We explain why cumulants are better suited for this job when using computer algebra. We provide concrete examples with distributions coming from exponential families and Diracs.