Positive margins and primary decomposition

Kahle, Thomas; Rauh, Johannes; Sullivant, Seth

doi:10.1216/jca-2014-6-2-173

Cited by 13 publications

(20 citation statements)

References 24 publications

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“…The answer to this question is negative. More than a year after first submission of the present paper, Kahle, Rauh, and Sullivant showed that the global Markov ideal of K 3,3 is not radical [20].…”

Section: Introductionmentioning

confidence: 73%

Multigraded commutative algebra of graph decompositions

2013

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The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same multigrading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We describe how to obtain generating sets of toric fiber products in non-zero codimension and discuss persistence of normality and primary decompositions under toric fiber products.Several applications are discussed, including (a) the construction of Markov bases of hierarchical models in many new cases, (b) a new proof of the quartic generation of binary graph models associated to K 4 -minor free graphs, and (c) the recursive computation of primary decompositions of conditional independence ideals. arXiv:1102.2601v5 [math.AC] 9 May 2014 Theorem 1.1. Let d i = 2 for all i ∈ V and let Γ be a graph with no K 4 minors. Then I Γ,d is generated by binomials of degrees two and four.Combining our techniques with results from [15], we can also make statements about the asymptotic behavior as the d i grow. For instance, let F ⊆ V be an independent set of Γ and consider I Γ,d as d i tend to infinity for i ∈ F , while the remaining d i are fixed. In this case, there is a bound M (Γ, d V \F ) for the degrees of elements in minimal generating sets of I Γ,d . Our techniques allow us to determine the values of M (Γ, d V \F ), which were previously known only for reducible models or when F is a singleton [17]. Here is a simple example of how to apply Theorem 5.15. Example 1.2. Let Γ = [12][13][24][34] be a four cycle, F = {1, 4}, and d {2,3} = (2, 2). The toric ideal I Γ,d is a codimension one toric fiber product and its minimal generating set consists of the following four types of binomials, written in tableau notation (a common J A⊥ ⊥B |C := p i A i An argument similar to that in Section 1.2 shows that this ideal is prime. For a collectionof CI-statements, the CI-ideal is the sum of the ideals of its statements:In statistics one is usually not interested in all of the variety of a CI-ideal, but only its intersection with the set of probability distributions. The following properties of CI-ideals imply well-known properties of conditional independence.Proposition 6.1. The following ideal containments hold:• J A⊥ ⊥B |C = J B⊥ ⊥A |C (symmetry);• J A⊥ ⊥B∪D |C ⊃ J A⊥ ⊥B |C (decomposition); • J A⊥ ⊥B∪D |C ⊃ J A⊥ ⊥B |C∪D (weak union).However, the contraction property does not hold algebraically since J A⊥ ⊥B |C∪D + J A⊥ ⊥D |C ⊇ J A⊥ ⊥B∪D |C .

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Section: Introductionmentioning

confidence: 73%

Multigraded commutative algebra of graph decompositions

2013

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“…A first step toward Problem 17.17 was developed in [KRS12]. There the authors study only the connectivity of F b as a function of the position of b in the cone Q + A. Additionally all ideals there are radical, and consequently the subtleties of mesoprimary decomposition play no role.…”

Section: Posets Of Mesoprimesmentioning

confidence: 99%

“…In algebraic statistics, decompositions of binomial ideals give insight into how a set of conditional independence statements among random variables can be realized [DSS09, HHH + ]. More generally, the connectivity of lattice point walks in polyhedra can be analyzed using decompositions of binomial ideals [DES98,KRS12]. These applications rely on decompositions of unital ideals-generated by monomials and differences of monomials-into unital ideals; these are mesoprimary decompositions.…”

Section: Introductionmentioning

confidence: 99%

Decompositions of commutative monoid congruences and binomial ideals

Kahle

Miller

2014

Algebra Number Theory

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Abstract. Primary decomposition of commutative monoid congruences is insensitive to certain features of primary decomposition in commutative rings. These features are captured by the more refined theory of mesoprimary decomposition of congruences, introduced here complete with witnesses and associated prime objects. The combinatorial theory of mesoprimary decomposition lifts to arbitrary binomial ideals in monoid algebras. The resulting binomial mesoprimary decomposition is a new type of intersection decomposition for binomial ideals that enjoys computational efficiency and independence from ground field hypotheses. Binomial primary decompositions are easily recovered from mesoprimary decomposition.

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“…It has been shown that in some cases, a smaller set of moves suffices under the condition of positive margins, for example, Chen et al49 and Hara et al50 demonstrate this in case of exact logistic regression. More recently, Kahle et al51 have studied the positive margin property and its generalization called the interior point property for graphical models by studying primary decompositions of conditional independence ideals.…”

Section: Computing Markov Basesmentioning

confidence: 99%

Conditional inference given partial information in contingency tables using Markov bases

Karwa

Slavković

2013

WIREs Computational Stats

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In this article, we review a Markov chain Monte Carlo (MCMC) algorithm for performing conditional inference in contingency tables in the presence of partial information using Markov bases, a key tool arising from the area known as algebraic statistics. We review applications of this algorithm to the problems of conditional exact tests, ecological inference, and disclosure limitation and illustrate how these problems fall naturally in the setting of inference with partial information. We also discuss some issues associated with computing Markov bases which are needed as an input to the algorithm. WIREs Comput Stat 2013, 5:207–218. doi: 10.1002/wics.1256 This article is categorized under: Data: Types and Structure > Traditional Statistical Data

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Positive margins and primary decomposition

Cited by 13 publications

References 24 publications

Multigraded commutative algebra of graph decompositions

Multigraded commutative algebra of graph decompositions

Decompositions of commutative monoid congruences and binomial ideals

Conditional inference given partial information in contingency tables using Markov bases

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