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Title: A linear-algebraic tool for conditional independence inference
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: A Family of Quasisymmetry Models
Date: 2015, 2015-06-11
Description: We present a one-parameter family of models for square contingency tables that interpolates between the classical quasisymmetry model and its...
Show moreWe present a one-parameter family of models for square contingency tables that interpolates between the classical quasisymmetry model and its Pearsonian analogue. Algebraically, this corresponds to deformations of toric ideals associated with graphs. Our discussion of the statistical issues centers around maximum likelihood estimation.
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Collection: Journal of Algebraic Statistics

Title: Markov degree of configurations defined by fibers of a configuration
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: On Polyhedral Approximations of Polytopes for Learning Bayesian Networks, AS2012 Special Volume, part 2: 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.
Date: 2013, 2013
Description: The motivation for this paper is the geometric approach to statistical learning Bayesiannetwork (BN) structures. We review three vector...
Show moreThe motivation for this paper is the geometric approach to statistical learning Bayesiannetwork (BN) structures. We review three vector encodings of BN structures. The first one has been used by Jaakkola et al. [9] and also by Cussens [4], the other two use special integral vectors formerly introduced, called imsets [18, 20]. The topic is the comparison of outer polyhedral approximations of the corresponding polytopes. We show how to transform the inequalities suggested by Jaakkola et al. [9] into the framework of imsets. The result of our comparison is the observation that the implicit polyhedral approximation of the standard imset polytope suggested in [21] gives a tighter approximation than the (transformed) explicit polyhedral approximation from [9]. As a consequence, we confirm a conjecture from [21] that the above-mentioned implicit polyhedral approximation of the standard imset polytope is an LP relaxation of that polytope. In the end, we review recent attempts to apply the methods of integer programming to learning BN structures and discuss the task of finding suitable explicit LP relaxation in the imset-based approach.
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Collection: Journal of Algebraic Statistics

Title: Higher Connectivity of Fiber Graphs of Gröbner Bases, AS2012 Special Volume, part 2: 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.
Date: 2013, 2013
Description: Fiber graphs of Gröbner bases from contingency tables are important in statistical hypothesis testing, where one studies random walks on these...
Show moreFiber graphs of Gröbner bases from contingency tables are important in statistical hypothesis testing, where one studies random walks on these graphs using the Metropolis-Hastings algorithm. The connectivity of the graphs has implications on how fast the algorithm converges. In this paper, we study a class of ber graphs with elementary combinatorial techniques and provide results that support a recent conjecture of Engström: the connectivity is given by the minimum vertex degree.
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Collection: Journal of Algebraic Statistics

Title: Betti Numbers of Cut Ideals of Trees, AS2012 Special Volume, part 2: 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.
Date: 2013, 2013
Description: Cut ideals, introduced by Sturmfels and Sullivant, are used in phylogenetics and algebraic statistics. We study the minimal free resolutions...
Show moreCut ideals, introduced by Sturmfels and Sullivant, are used in phylogenetics and algebraic statistics. We study the minimal free resolutions of cut ideals of tree graphs. By employing basic methods from topological combinatorics, we obtain upper bounds for the Betti numbers of this type of ideals. These take the form of simple formulas on the number of vertices, which arise from the enumeration of induced subgraphs of certain incomparability graphs associated to the edge sets of trees.
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Collection: Journal of Algebraic Statistics

Title: Tying Up Loose Strands: Defining Equations of the Strand Symmetric Model
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: The maximum likelihood degree of Fermat hypersurfaces
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: On the Connectivity of Fiber Graphs
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

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