Markov chains, quotient ideals and connectivity with positive margins

Chen, Yuguo; Dinwoodie, Ian H.; Yoshida, Ruriko

doi:10.1017/cbo9780511642401.006

Cited by 14 publications

(23 citation statements)

References 25 publications

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“…Rapallo and Yoshida48 study Markov subbases for bounded contingency tables for the model of independence. 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: 98%

“…A related basis called lattice basis works by allowing the Markov chain to have tables with values such as −1, for example, see Refs 49, 52, and also Ref 53 for advantages and disadvantages of lattice bases in comparison to Markov bases. This idea of going outside the original fiber that has only nonnegative tables, has been used in cases where Markov bases are difficult to compute for the original fiber ℱ 0 but easier for an enlarged fiber ℱ 1 by using the Lawrence lifting.…”

Section: Computing Markov Basesmentioning

confidence: 99%

See 1 more Smart Citation

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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Section: Computing Markov Basesmentioning

confidence: 98%

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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“…[11], [2], [3]), this result is closely related to connectivity of a specific fiber by a subset of a Markov basis. See [4], [1], [12] for relevant results. Since a Markov basis for the whole configuration of elementary imsets is very complicated ( [5]), it is remarkable that F A,B | C is connected by the two-by-two basic relations.…”

Section: Theorem 1 Every F Ab | C Is Connected By Two-by-two Basic mentioning

confidence: 99%

Properties of semi-elementary imsets as sums of elemen- tary imsets

Kashimura¹,

Sei²,

Takemura³

et al. 2011

J Alg Stat

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We study properties of semi-elementary imsets and elementary imsets introduced by Studený [10]. The rules of the semi-graphoid axiom (decomposition, weak union and contraction) for conditional independence statements can be translated into a simple identity among three semi-elementary imsets. By recursively applying the identity, any semi-elementary imset can be written as a sum of elementary imsets, which we call a representation of the semi-elementary imset. A semi-elementary imset has many representations. We study properties of the set of possible representations of a semi-elementary imset and prove that all representations are connected by relations among four elementary imsets.2000 Mathematics Subject Classifications: 62F15, 90C10

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“…For graphical models there is a canonical set of "simple" moves, which correspond to the global Markov conditional independence statements. It has been observed that for some models, if a contingency table u has strictly positive margins, then these simple moves connect the fiber of u [6]. Similarly, for the no-three-way interaction model Bunea and Besag proved a positive margins property for a set of "simple" moves [5].…”

Section: Introductionmentioning

confidence: 98%

Positive margins and primary decomposition

Kahle¹,

Rauh²,

Sullivant³

2014

J. Commut. Algebra

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We study random walks on contingency tables with fixed marginals, corresponding to a (log-linear) hierarchical model. If the set of allowed moves is not a Markov basis, then there exist tables with the same marginals that are not connected. We study linear conditions on the values of the marginals that ensure that all tables in a given fiber are connected. We show that many graphical models have the positive margins property, which says that all fibers with strictly positive marginals are connected by the quadratic moves that correspond to conditional independence statements. The property persists under natural operations such as gluing along cliques, but we also construct examples of graphical models not enjoying this property. We also provide a negative answer to a question of Engstr\"om, Kahle, and Sullivant by demonstrating that the global Markov ideal of the complete bipartite graph K_(3,3) is not radical. Our analysis of the positive margins property depends on computing the primary decomposition of the associated conditional independence ideal. The main technical results of the paper are primary decompositions of the conditional independence ideals of graphical models of the $N$-cycle and the complete bipartite graph $K_(2,N-2)$, with various restrictions on the size of the nodes.Comment: 26 pages, 3 figures, v2: various small improvements, v3: added K_(3,3) as an example of a non-radical global Markov ideal + small improvement

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Markov chains, quotient ideals and connectivity with positive margins

Cited by 14 publications

References 25 publications

Conditional inference given partial information in contingency tables using Markov bases

Conditional inference given partial information in contingency tables using Markov bases

Properties of semi-elementary imsets as sums of elemen- tary imsets

Positive margins and primary decomposition

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