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Title: The maximum likelihood degree of Fermat hypersurfaces
Creator: Agostini, Daniele, Alberelli, Davide, Grande, Francesco, Lella, Paolo
Date: 2015, 2015-11-09
Description: We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in...
Show moreWe study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section.
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Collection: Journal of Algebraic Statistics

Title: 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.
Creator: Améndola, Carlos, Faugère, Jean-Charles, Sturmfels, Bernd
Date: 2016, 2016-07-12
Description: 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.
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Collection: Journal of Algebraic Statistics

Title: On multivariable cumulant polynomial sequences with applications, 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.
Creator: Di Nardo, Elvira
Date: 2016, 2016-07-12
Description: 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.
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Collection: Journal of Algebraic Statistics

Title: Algebraic geometry of Poisson regression, 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.
Creator: Kahle, Thomas, Oelbermann, Kai-Friederike, Schwabe, Rainer
Date: 2016, 2016-07-12
Description: 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.
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Collection: Journal of Algebraic Statistics

Title: A linear-algebraic tool for conditional independence inference
Creator: Tanaka, Kentaro, Studeny, Milan, Takemura, Akimichi, Sei, Tomonari
Date: 2015, 2015-11-09
Description: 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.
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Collection: Journal of Algebraic Statistics

Title: Mode Poset Probability Polytopes, 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.
Creator: Montúfar, Guido, Rauh, Johannes
Date: 2016, 2016-07-12
Description: 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.
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Collection: Journal of Algebraic Statistics

Title: Technology News, April 01, 1949
Creator: Illinois Institute of Technology
Date: 1949-04-01, 1949-04-01
Collection: Technology News Microfilm collection, 1928-1981

Title: Ken Isaacs with the Knowledge Box, Illinois Institute of Technology, Chicago, Illinois, 1962
Creator: Illinois Institute of Technology. Institute of Design
Date: 1962
Description: Photograph of Institute of Design visiting lecturer Ken Isaacs and the Knowledge Box.
Collection: Institute of Design records, 1937-ca. 1962

Title: Ken Isaacs with the Knowledge Box, Illinois Institute of Technology, Chicago, Illinois, 1962
Creator: Illinois Institute of Technology. Institute of Design
Date: 1962
Description: Photograph of Institute of Design visiting lecturer Ken Isaacs and the Knowledge Box.
Collection: Institute of Design records, 1937-ca. 1962

Title: Ken Isaacs with the Knowledge Box, Illinois Institute of Technology, Chicago, Illinois, 1962
Creator: Illinois Institute of Technology. Institute of Design
Date: 1962
Description: Photograph of Institute of Design visiting lecturer Ken Isaacs and the Knowledge Box.
Collection: Institute of Design records, 1937-ca. 1962

Title: New Bauhaus scrapbook, 1936-1938
Date: 1936-1938
Description: 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.
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Collection: Institute of Design records, 1937-ca. 1962