Moment Varieties of Gaussian Mixtures

Moment Varieties of Gaussian Mixtures AS2015 Special Issue articles: This issue includes a series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics Conference held in Genoa, Italy, in June 2015. Special issue guest editors: Piotr Zwiernik and Fabio Rapallo. Améndola, Carlos Creator Faculty carlos.amendola@tum.de Faugère, Jean-Charles Creator Sturmfels, Bernd Creator MATH / Applied Mathematics Illinois Institute of Technology Affiliated department Algebraic statistics method of moments mixture model normal distribution secant varieties 2016 2016-07-12 The points of a moment variety are the vectors of all moments up to some order, for a given family of probability distributions. We study the moment varieties for mixtures of multivariate Gaussians. Following up on Pearson’s classical work from 1894, we apply current tools from computational algebra to recover the parameters from the moments. Our moment varieties extend objects familiar to algebraic geometers. For instance, the secant varieties of Veronese varieties are the loci obtained by setting all covariance matrices to zero. We compute the ideals of the 5-dimensional moment varieties representing mixtures of two univariate Gaussians, and we offer a comparison to the maximum likelihood approach. Article