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 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, 20160712
 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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 Journal of Algebraic Statistics
 Title
 Lcumulants, Lcumulant 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 Lcumulants. This generalization...
Show moreFocusing on the discrete probabilistic setting we generalize the combinatorial definition of cumulants to Lcumulants. This generalization keeps all the desired properties of the classical cumulants like semiinvariance and vanishing for independent blocks of random variables. These properties make Lcumulants 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 cumulantlike 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 Lcumulants can be used in the analysis of certain classical algebraic varieties.
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 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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 Journal of Algebraic Statistics
 Title
 Maximal Length Projections in Group Algebras with Applications to Linear Rank Tests of Uniformity
 Creator
 Bargagliotti, Anna E., Orrison, Michael
 Description

Let G be a finite group, let CG be the complex group algebra of G, and let p ∈ CG. In this paper, we show how to construct submodules S of CG...
Show moreLet G be a finite group, let CG be the complex group algebra of G, and let p ∈ CG. In this paper, we show how to construct submodules S of CG of a fixed dimension with the property that the orthogonal projection of p onto S has maximal length. We then provide an example of how such submodules for the symmetric group Sn can be used to create new linear rank tests of uniformity in statistics for survey data that arises when respondents are asked to give a complete ranking of n items.
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 Journal of Algebraic Statistics
 Title
 Connectivity for 3 x 3 x K contingency tables
 Creator
 Sumi, Toshio, 2012
 Description

We consider an exact sequential conditional test for threeway conditional test of no interaction. At each time τ, the test uses as the...
Show moreWe consider an exact sequential conditional test for threeway conditional test of no interaction. At each time τ, the test uses as the conditional inference frame the set F(Hτ) of all tables with the same three twoway marginal tables as the obtained table Hτ . For 3 × 3 × K tables, we propose a method to construct F(Hτ) from F(Hτ−1). This enables us to perform efficiently the sequential exact conditional test. The subset Sτ of F (Hτ ) consisting of s + Hτ − Hτ −1 for s ∈ F(Hτ−1) contains Hτ , where the operations + and − are defined elementwise. Our argument is based on the minimal Markov basis for 3 × 3 × K contingency tables and we give a minimal subset M of some Markov basis which has the property that F (Hτ ) = {s − m  s ∈ Sτ , m ∈ M}.
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 Journal of Algebraic Statistics
 Title
 Matrix Completion for the Independence Model
 Creator
 Kubjas, Kaie, Rosen, Zvi
 Description

We investigate the problem of completing partial matrices to rankone matrices in the standard simplex ∆mn−1. The motivation for studying this...
Show moreWe investigate the problem of completing partial matrices to rankone matrices in the standard simplex ∆mn−1. The motivation for studying this problem comes from statistics: A lack of eligible completion can provide a falsification test for partial observations to come from the independence model. For each pattern of specified entries, we give equations and inequalities which are satisfied if and only if an eligible completion exists. We also describe the set of valid completions, and we optimize over this set.
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 Journal of Algebraic Statistics
 Title
 One example of general unidentifiable tensors
 Creator
 Chiantini, Luca, Mella, Massimiliano, Ottaviani, Giorgio
 Description

Abstract. Theidentifiabilityofparametersinaprobabilisticmodelisacrucialnotioninstatistical inference. We prove that a general tensor of rank 8...
Show moreAbstract. Theidentifiabilityofparametersinaprobabilisticmodelisacrucialnotioninstatistical inference. We prove that a general tensor of rank 8 in C3 ⊗ C6 ⊗ C6 has at least 6 decompositions as sum of simple tensors, so it is not 8identifiable. This is the highest known example of balanced tensors of dimension 3, which are not kidentifiable, when k is smaller than the generic rank.
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 Journal of Algebraic Statistics
 Title
 Hilbert Polynomial of the Kimura 3Parameter Model, 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
 Kubjas, Kaie, inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July
 Description

In [2] Buczyn ́ska and Wi ́sniewski showed that the Hilbert polynomial of the algebraic variety associated to the JukesCantor binary model on...
Show moreIn [2] Buczyn ́ska and Wi ́sniewski showed that the Hilbert polynomial of the algebraic variety associated to the JukesCantor binary model on a trivalent tree depends only on the number of leaves of the tree and not on its shape. We ask if this can be generalized to other groupbased models. The JukesCantor binary model has Z2 as the underlying group. We consider the Kimura 3parameter model with Z2 × Z2 as the underlying group. We show that the generalization of the statement about the Hilbert polynomials to the Kimura 3parameter model is not possible as the Hilbert polynomial depends on the shape of a trivalent tree.
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 Journal of Algebraic Statistics
 Title
 Stephen Fienberg's influence on algebraic statistics, Special Volume in honor of memory of S.E.Fienberg
 Creator
 Petrović, Sonja, Slavkovic, Aleksandra, Yoshida, Ruriko
 Date
 2019, 20190412
 Description

Stephen (Steve) E. Fienberg (19422016) was an eminent statistician, whose impact on research, education and the practice of statistics, and...
Show moreStephen (Steve) E. Fienberg (19422016) was an eminent statistician, whose impact on research, education and the practice of statistics, and many other fields is astonishing in its breadth. He was a visionary when it came to linking many different areas to address real scientific issues. He professed the importance of statistics in many disciplines, but recognized that true interdisciplinary work requires joining of the expertise across different areas, and it is in this spirit that he helped steer algebraic statistics toward becoming a thriving subject. Many of his favorite topics in the area are covered in this special issue. We are grateful to all authors for contributing to this volume to honor him and his influence on the field. During the preparation of this issue, we learned about the tragic killing of his widow, Joyce Fienberg, during the Tree of Life Synagogue massacre in Pittsburgh, PA on October 27, 2018. This issue is dedicated to their memory.
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 Journal of Algebraic Statistics