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... Show moreThe 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. Show less
A new family of polynomials, called cumulant polynomial sequence, and its extension to the multivariate case is introduced relying on a purely... Show moreA new family of polynomials, called cumulant polynomial sequence, and its extension to the multivariate case is introduced relying on a purely symbolic combinatorial method. The coefficients are cumulants, but depending on what is plugged in the indeterminates, moment sequences can be recovered as well. The main tool is a formal generalization of random sums, when a not necessarily integer-valued multivariate random index is considered. Applications are given within parameter estimations, L ?evy processes and random matrices and, more generally, problems involving multivariate functions. The connection between exponential models and multivariable Sheffer polynomial sequences offers a different viewpoint in employing the method. Some open problems end the paper. Show less
Designing experiments for generalized linear models is difficult because optimal designs depend on unknown parameters. Here we investigate... Show moreDesigning experiments for generalized linear models is difficult because optimal designs depend on unknown parameters. Here we investigate local optimality. We propose to study for a given design its region of optimality in parameter space. Often these regions are semi-algebraic and feature interesting symmetries. We demonstrate this with the Rasch Poisson counts model. For any given interaction order between the explanatory variables we give a characterization of the regions of optimality of a special saturated design. This extends known results from the case of no interaction. We also give an algebraic and geometric perspective on optimality of experimental designs for the Rasch Poisson counts model using polyhedral and spectrahedral geometry. Show less
In this note, we propose a new linear-algebraic method for the implication problem among conditional independence statements, which is... Show moreIn this note, we propose a new linear-algebraic method for the implication problem among conditional independence statements, which is inspired by the factorization characterization of conditional independence. First, we give a criterion in the case of a discrete strictly positive density and relate it to an earlier linear-algebraic approach. Then, we extend the method to the case of a discrete density that need not be strictly positive. Finally, we provide a computational result in the case of six variables. Show less
A mode of a probability distribution is an elementary event that has more probability mass than each of its direct neighbors, with respect to... Show moreA mode of a probability distribution is an elementary event that has more probability mass than each of its direct neighbors, with respect to some vicinity structure on the set of elementary events. The mode inequalities cut out a polytope from the simplex of probability distributions. Related to this is the concept of strong modes. A strong mode is an elementary event that has more probability mass than all its direct neighbors together. The set of probability distributions with a given set of strong modes is again a polytope. We study the vertices, the facets, and the volume of such polytopes depending on the sets of (strong) modes and the vicinity structures. Show less
Photograph of Illinois Tech fraternity and sorority members in a bed race during Greek Week. Photographer unknown. Date of photograph is... Show morePhotograph of Illinois Tech fraternity and sorority members in a bed race during Greek Week. Photographer unknown. Date of photograph is unknown. Date listed is approximate. Show less
Thirty-seven page scrapbook with hand made fabric covered box with lower case letter b (for Bauhaus) on font in silver metal. Scrapbook (16 x... Show moreThirty-seven page scrapbook with hand made fabric covered box with lower case letter b (for Bauhaus) on font in silver metal. Scrapbook (16 x 16”) has black plastic covers, spiral bound with manila card stock pages. Includes news clippings and tear sheets from periodicals about the opening of The New Bauhaus as well as the academic catalogue and program announcements, and a letter from Walter Gropius (5/18/1937) recommending Laszlo Moholy-Nagy as director. Creator unknown, assumed to be affiliated with the New Bauhaus. Show less