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Title: On the Connectivity of Fiber Graphs
Creator: Hemmecke, Raymond, Windisch, Tobias
Date: 2015, 2015-06-11
Description: We consider the connectivity of fiber graphs with respect to Gröbner basis and Graver basis moves. First, we present a sequence of fiber...
Show moreWe consider the connectivity of fiber graphs with respect to Gröbner basis and Graver basis moves. First, we present a sequence of fiber graphs using moves from a Gröbner basis and prove that their edge-connectivity is lowest possible and can have an arbitrarily large distance from the minimal degree. We then show that graph-theoretic properties of fiber graphs do not depend on the size of the right-hand side. This provides a counterexample to a conjecture of Engström on the node-connectivity of fiber graphs. Our main result shows that the edge-connectivity in all fiber graphs of this counterexample is best possible if we use moves from Graver basis instead.
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Collection: Journal of Algebraic Statistics

Title: The precision space of interpolatory cubature formulæ
Creator: Fassino, Claudia, Riccomagno, Eva
Date: 2015, 2015-06-11
Description: Methods from Commutative Algebra and Numerical Analysis are combined to address a problem common to many disciplines: the estimation of the...
Show moreMethods from Commutative Algebra and Numerical Analysis are combined to address a problem common to many disciplines: the estimation of the expected value of a polynomial of a random vector using a linear combination of a finite number of its values. In this work we remark on the error estimation in cubature formulæ for polynomial functions and introduce the notion of a precision space for a cubature rule.
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Collection: Journal of Algebraic Statistics

Title: The degeneration of the Grassmannian into a toric variety and the calculation of the eigenspaces of a torus action
Creator: Witaszek, Jakub
Date: 2015, 2015-06-11
Description: Using the method of degenerating a Grassmannian into a toric variety, we calculate formulas for the dimensions of the eigenspaces of the...
Show moreUsing the method of degenerating a Grassmannian into a toric variety, we calculate formulas for the dimensions of the eigenspaces of the action of an n-dimensional torus on a Grassmannian of planes in an n-dimensional space.
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Collection: Journal of Algebraic Statistics

Title: Varieties with maximum likelihood degree one
Creator: Huh, June
Date: 2014, 2014-04-30
Description: We show that algebraic varieties with maximum likelihood degree one are exactly the images of reduced A-discriminantal varieties under...
Show moreWe show that algebraic varieties with maximum likelihood degree one are exactly the images of reduced A-discriminantal varieties under monomial maps with finite fibers. The maximum likelihood estimator corresponding to such a variety is Kapranov’s Horn uniformization. This extends Kapranov’s characterization of A-discriminantal hypersurfaces to varieties of arbitrary codimension.
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Collection: Journal of Algebraic Statistics

Title: Tying Up Loose Strands: Defining Equations of the Strand Symmetric Model
Creator: Long, Colby, Sullivant, Seth
Date: 2015, 2015-06-11
Description: The strand symmetric model is a phylogenetic model designed to reflect the symmetry inherent in the double-stranded structure of DNA. We show...
Show moreThe strand symmetric model is a phylogenetic model designed to reflect the symmetry inherent in the double-stranded structure of DNA. We show that the set of known phylogenetic invariants for the general strand symmetric model of the three leaf claw tree entirely defines the ideal. This knowledge allows one to determine the vanishing ideal of the general strand symmetric model of any trivalent tree. Our proof of the main result is computational. We use the fact that the Zariski closure of the strand symmetric model is the secant variety of a toric variety to compute the dimension of the variety. We then show that the known equations generate a prime ideal of the correct dimension using elimination theory.
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Collection: Journal of Algebraic Statistics

Title: On Exchangeability in Network Models, Special Volume in honor of memory of S.E.Fienberg
Creator: Lauritzen, Steffen, Rinaldo, Alessandro, Sadeghi, Kayvan
Date: 2019, 2019-04-12
Description: We derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric...
Show moreWe derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric and graph-theoretic concepts. Our results elucidate some of the key differences, and their implications, between statistical network models that are finitely exchangeable and models that define a consistent sequence of probability distributions on graphs of increasing size.
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Collection: Journal of Algebraic Statistics

Title: Maximum Likelihood Estimation of the Latent Class Model through Model Boundary Decomposition, Special Volume in honor of memory of S.E.Fienberg
Creator: Elizabeth S. Allman, Baños Cervantes, Hector, Evans, Robin, Hosten, Serkan, Kubjas, Kaie, Lemke, Daniel, Rhodes, John, Zwiernik, Piotr
Date: 2019, 2019-04-12
Description: The Expectation-Maximization (EM) algorithm is routinely used for maximum likelihood estimation in latent class analysis. However, the EM...
Show moreThe Expectation-Maximization (EM) algorithm is routinely used for maximum likelihood estimation in latent class analysis. However, the EM algorithm comes with no global guarantees of reaching the global optimum. We study the geometry of the latent class model in order to understand the behavior of the maximum likelihood estimator. In particular, we characterize the boundary stratification of the binary latent class model with a binary hidden variable. For small models, such as for three binary observed variables, we show that this stratification allows exact computation of the maximum likelihood estimator. In this case we use simulations to study the maximum likelihood estimation attraction basins of the various strata and performance of the EM algorithm. Our theoretical study is complemented with a careful analysis of the EM fixed point ideal which provides an alternative method of studying the boundary stratification and maximizing the likelihood function. In particular, we compute the minimal primes of this ideal in the case of a binary latent class model with a binary or ternary hidden random variable.
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Collection: Journal of Algebraic Statistics

Title: Inference for Ordinal Log-Linear Models Based on Algebraic Statistics, Special Volume in honor of memory of S.E.Fienberg
Creator: Pham, Thi Mui, Kateri, Maria
Date: 2019, 2019-04-12
Description: Tools of algebraic statistics combined with MCMC algorithms have been used in contingency table analysis for model selection and model fit...
Show moreTools of algebraic statistics combined with MCMC algorithms have been used in contingency table analysis for model selection and model fit testing of log-linear models. However, this approach has not been considered so far for association models, which are special log-linear models for tables with ordinal classification variables. The simplest association model for two-way tables, the uniform (U) association model, has just one parameter more than the independence model and is applicable when both classification variables are ordinal. Less parsimonious are the row (R) and column (C) effect association models, appropriate when at least one of the classification variables is ordinal. Association models have been extended for multidimensional contingency tables as well. Here, we adjust algebraic methods for association models analysis and investigate their eligibility, focusing mainly on two-way tables. They are implemented in the statistical software R and illustrated on real data tables. Finally the algebraic model fit and selection procedure is assessed and compared to the asymptotic approach in terms of a simulation study.
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Collection: Journal of Algebraic Statistics

Title: Exact tests to compare contingency tables under quasi-independence and quasi-symmetry, Special Volume in honor of memory of S.E.Fienberg
Creator: Bocci, Christiano, Rapallo, Fabio
Date: 2019, 2019-04-12
Description: In this work we define log-linear models to compare several square contingency tables under the quasi-independence or the quasi-symmetry model...
Show moreIn this work we define log-linear models to compare several square contingency tables under the quasi-independence or the quasi-symmetry model, and the relevant Markov bases are theoretically characterized. Through Markov bases, an exact test to evaluate if two or more tables fit a common model is introduced. Two real-data examples illustrate the use of tehse models in different fields of applications.
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Collection: Journal of Algebraic Statistics

Title: Detecting epistasis via Markov bases
Creator: Malaspinas, Anna-Sapfo, Uhler, Caroline
Date: 2011, 2011
Description: Rapid research progress in genotyping techniques have allowed large genome-wide association studies. Existing methods often focus on...
Show moreRapid research progress in genotyping techniques have allowed large genome-wide association studies. Existing methods often focus on determining associations between single loci and a specific phenotype. However, a particular phenotype is usually the result of complex relationships between multiple loci and the environment. In this paper, we describe a two-stage method for detecting epistasis by combining the traditionally used single-locus search with a search for multiway interactions. Our method is based on an extended version of Fisher’s exact test. To perform this test, a Markov chain is constructed on the space of multidimensional contingency tables using the elements of a Markov basis as moves. We test our method on simulated data and compare it to a two-stage logistic regression method and to a fully Bayesian method, showing that we are able to detect the interacting loci when other methods fail to do so. Finally, we apply our method to a genome-wide data set consisting of 685 dogs and identify epistasis associated with canine hair length for four pairs of single nucleotide polymorphisms (SNPs).
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Collection: Journal of Algebraic Statistics

Title: Open Problems on Connectivity of Fibers with Positive Margins in Multi-dimensional Contingency Tables
Creator: Yoshida, Ruriko
Date: 2010, 2010
Description: Diaconis-Sturmfels developed an algorithm for sampling from conditional distributions for a statistical model of discrete exponential families...
Show moreDiaconis-Sturmfels developed an algorithm for sampling from conditional distributions for a statistical model of discrete exponential families, based on the algebraic theory of toric ideals. This algorithm is applied to categorical data analysis through the notion of Markov bases. Initiated with its application to Markov chain Monte Carlo approach for testing statistical fitting of the given model, many researchers have extensively studied the structure of Markov bases for models in computational algebraic statistics. In the Markov chain Monte Carlo approach for testing statistical fitting of the given model, a Markov basis is a set of moves connecting all contingency tables satisfying the given margins. Despite the computational advances, there are applied problems where one may never be able to compute a Markov basis. In general, the number of elements in a minimal Markov basis for a model can be exponentially many. Thus, it is important to compute a reduced number of moves which connect all tables instead of computing a Markov basis. In some cases, such as logistic regression, positive margins are shown to allow a set of Markov connecting moves that are much simpler than the full Markov basis. Such a set is called a Markov subbasis with assumption of positive margins. In this paper we summarize some computations of and open problems on Markov subbases for contingency tables with assumption of positive margins under specific models as well as develop algebraic methods for studying connectivity of Markov moves with margin positivity to develop Markov sampling methods for exact conditional inference in statistical models where the Markov basis is hard to compute.
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Collection: Journal of Algebraic Statistics

Title: Uncovering Proximity of Chromosome Territories using Classical Algebraic Statistics
Creator: Arsuaga, Javier, Heskia, Ido, Hosten, Serkan, Maskalevich, Tatiana
Date: 2015, 2015-11-09
Description: Exchange type chromosome aberrations (ETCAs) are rearrangements of the genome that occur when chromosomes break and the resulting fragments...
Show moreExchange type chromosome aberrations (ETCAs) are rearrangements of the genome that occur when chromosomes break and the resulting fragments rejoin with fragments from other chromosomes or from other regions within the same chromosome. ETCAs are commonly observed in cancer cells and in cells exposed to radiation. The frequency of these chromosome rearrangements is correlated with their spatial proximity, therefore it can be used to infer the three dimensional organization of the genome. Extracting statistical significance of spatial proximity from cancer and radiation data has remained somewhat elusive because of the sparsity of the data. We here propose a new approach to study the three dimensional organization of the genome using algebraic statistics. We test our method on a published data set of irradiated human blood lymphocyte cells. We provide a rigorous method for testing the overall organization of the genome, and in agreement with previous results we find a random relative positioning of chromosomes with the exception of the chromosome pairs {1,22} and {13,14} that have a significantly larger number of ETCAs than the rest of the chromosome pairs suggesting their spatial proximity. We conclude that algebraic methods can successfully be used to analyze genetic data and have potential applications to larger and more complex data sets.
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Collection: Journal of Algebraic Statistics

Title: Markov degree of configurations defined by fibers of a configuration
Creator: Koyama, Takayuki, Ogawa, Mitsunori, Takemura, Akimichi
Date: 2015, 2015-11-09
Description: We consider a series of configurations defined by fibers of a given base configuration. We prove that Markov degree of the configurations is...
Show moreWe consider a series of configurations defined by fibers of a given base configuration. We prove that Markov degree of the configurations is bounded from above by the Markov complexity of the base configuration. As important examples of base configurations we consider incidence matrices of graphs and study the maximum Markov degree of configurations defined by fibers of the incidence matrices. In particular we give a proof that the Markov degree for two-way transportation polytopes is three.
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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: L-cumulants, L-cumulant embeddings and algebraic statistics, AS2012 Special Volume, part 1: This issue includes a second series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July 2012.
Creator: Zwiernik, Piotr, AS2012 Special Volume, part 1: This issue includes a second series of papers from talks, posters and collaborations resulting from and
Description: Focusing on the discrete probabilistic setting we generalize the combinatorial definition of cumulants to L-cumulants. This generalization...
Show moreFocusing on the discrete probabilistic setting we generalize the combinatorial definition of cumulants to L-cumulants. This generalization keeps all the desired properties of the classical cumulants like semi-invariance and vanishing for independent blocks of random variables. These properties make L-cumulants useful for the algebraic analysis of statistical models. We illustrate this for general Markov models and hidden Markov processes in the case when the hidden process is binary. The main motivation of this work is to understand cumulant-like coordinates in alge- braic statistics and to give a more insightful explanation why tree cumulants give such an elegant description of binary hidden tree models. Moreover, we argue that L-cumulants can be used in the analysis of certain classical algebraic varieties.
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Collection: Journal of Algebraic Statistics

Title: Generalized Fréchet Bounds for Cell Entries in Multidimensional Contingency Tables, Special Volume in honor of memory of S.E.Fienberg
Creator: Richards, Donald, Uhler, Caroline
Description: We consider the lattice, L, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and...
Show moreWe consider the lattice, L, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, n(·), on L. We derive from the supermodularity of n(·) some generalized Fr ́echet inequalities comple- menting and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from n(·), and we remark on the connection between supermodularity and some correlation inequalities for probability distributions on lattices. We also apply an inequal- ity of Ky Fan to derive a new approach to Fr ́echet inequalities for multidimensional contingency tables.
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Collection: Journal of Algebraic Statistics

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