Markov Bases in Algebraic Statistics

Aoki, Satoshi; Hara, Hisayuki; Takemura, Akimichi

doi:10.1007/978-1-4614-3719-2

Cited by 76 publications

(101 citation statements)

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“…In the algebraic statistics setting, we are interested in a toric ideal representation where the marginals form a simplicial complex,24 and the appropriate design matrix T completely specifies the model. For a very nice textbook introduction to exact tests using Markov bases, examples and their applications in two‐way and k ‐way tables, see chapters 1 and 2 of Ref 5 and chapter 1 of Ref 14. In the Conclusion we also point to other methods for sampling from the fiber that can be used for computing p ‐values.…”

Section: Exact Tests In Contingency Tablesmentioning

confidence: 99%

“…Thus, Markov bases can be used for building a connected Markov chain. The connection between other bases of an integer lattice and Markov bases, in relation to the study of contingency tables, are explained in more detail, for example, in Chapter 1 of Ref 14 and in Chapter 4 of Ref 5.…”

Section: Conditional Inference With Markov Basesmentioning

confidence: 99%

“…In their seminal article on sampling from discrete exponential family distributions, Diaconis and Sturmfels4 show that such moves can be obtained generating sets of toric ideals and are called Markov bases. A comprehensive study of Markov bases and their use in algebraic statistics has been most recently captured in a book by Aoki et al5; for more on algebraic statistics, see Refs 6 and 7.…”

Section: Introductionmentioning

confidence: 99%

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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: Exact Tests In Contingency Tablesmentioning

confidence: 99%

Section: Conditional Inference With Markov Basesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Conditional inference given partial information in contingency tables using Markov bases

Karwa

Slavković

2013

WIREs Computational Stats

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“…Many special cases have been dealt with in algebraic statistics, though. See [2] and references therein.…”

Section: The Rasch Poisson Counts Modelmentioning

confidence: 99%

Algebraic geometry of Poisson regression

Kahle¹,

Oelbermann²,

Schwabe³

2016

J Alg Stat

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Abstract. Designing 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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“…Additionally, Graver bases and Markov bases can be easily computed for unimodular matrices, and efficient generation of Markov basis elements is an important problem in algebraic statistics. For more about Markov bases and their uses, see the books [2,10,12].…”

Section: Introductionmentioning

confidence: 99%

Unimodular hierarchical models and their Graver bases

Bernstein¹,

O’Neill²

2017

J Alg Stat

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Abstract. Given a simplicial complex whose vertices are labeled with positive integers, one can associate a vector configuration whose corresponding toric variety is the Zariski closure of a hierarchical model. We classify all the vertex-weighted simplicial complexes that give rise to unimodular vector configurations. We also provide a combinatorial characterization of their Graver bases. 2000 Mathematics Subject Classifications: 13P25, 52C07

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Markov Bases in Algebraic Statistics

Cited by 76 publications

References 0 publications

Conditional inference given partial information in contingency tables using Markov bases

Conditional inference given partial information in contingency tables using Markov bases

Algebraic geometry of Poisson regression

Unimodular hierarchical models and their Graver bases

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