Moment ideals of local Dirac mixtures

Moments are quantities that measure the shape of statistical or stochastic objects and have recently been studied from a more algebraic and combinatorial point of view. We start this talk by introducing local mixtures of Dirac measures and their moment ideals. We explain how taking local mixtures of the measures on the statistical side corresponds to the tangents of the algebraic varieties on the geometric one. We then use techniques from commutative algebra to find generators of the moment ideal for the Dirac measures and use deduce the generators of the moment ideal for the related Pareto distribution from those. Furthermore, we apply elimination theory and Prony's method in order to do parameter estimation, and showcase our results with an application in signal processing.

A main goal is to highlight the natural connections between algebraic statistics, geometry, combinatorics and applications in analysis throughout the talk.