scholarly journals What is... a Markov Basis?

2019 ◽  
Vol 66 (07) ◽  
pp. 1
Author(s):  
Sonja Petrović
Keyword(s):  
Markov Basis

scholarly journals Analysis of an Agricluture Data Using Markov Basis for Independent Model

2020 ◽  
Vol 1530 ◽  
pp. 012071
Author(s):  
Maysaa Jalil Mohammed ◽  
Ali Talib Mohammed
Keyword(s):  
Markov Basis ◽  
Independent Model

Generated Markov Basis from HHB

2020 ◽  
Vol 18 (7) ◽  
pp. 49-60
Author(s):  
Husein Hadi Abbas ◽  
Huda Qusay Hashim
Keyword(s):  
Markov Basis

scholarly journals Connectivity for 3 x 3 x K contingency tables

2011 ◽  
Vol 2 (1) ◽  
Author(s):  
Toshio Sumi ◽  
Toshio Sakata
Keyword(s):  
Contingency Tables ◽  
Conditional Inference ◽  
Markov Basis ◽  
Conditional Test ◽  
Marginal Tables

We consider an exact sequential conditional test for three-way conditional test of nointeraction. At each time , the test uses as the conditional inference frame the set F(H ) of alltables with the same three two-way marginal tables as the obtained table H . For 33K tables,we propose a method to construct F(H ) from F(H􀀀1). This enables us to perform ecientlythe sequential exact conditional test. The subset S of F(H ) consisting of s + H 􀀀 H􀀀1 fors 2 F(H􀀀1) contains H , where the operations + and 􀀀 are dened elementwise. Our argumentis based on the minimal Markov basis for 3 3 K contingency tables and we give a minimalsubset M of some Markov basis which has the property that F(H ) = fs 􀀀 m j s 2 S ;m 2 Mg.

Geometrically Aware Dynamic Markov Bases for Statistical Linear Inverse Problems

Biometrika ◽  
2020 ◽  
Author(s):  
M L Hazelton ◽  
M R Mcveagh ◽  
B Van Brunt
Keyword(s):  
Random Walk ◽  
Inverse Problems ◽  
Count Data ◽  
Latent Variable ◽  
Convex Polytope ◽  
Matrix Inversion ◽  
Lattice Points ◽  
Markov Basis ◽  
Linear Inverse Problems ◽  
Data Inference

Abstract For statistical linear inverse problems involving count data, inference typically requires sampling a latent variable with conditional support comprising the lattice points in a convex polytope. Irreducibility of random walk samplers is guaranteed only if a sufficiently rich array of sampling directions is available. In principle this can be achieved by finding a Markov basis of moves ab initio, but in practice doing so may be computationally infeasible. What is more, the use of a full Markov basis can lead to very poor mixing. It is far simpler to find a lattice basis of moves, which can be tailored to the overall geometry of the polytope. However, a single lattice basis generally does not connect all points in the polytope. In response, we propose a dynamic lattice basis sampler. This sampler can access a sufficient variety of sampling directions to guarantee irreducibility, but also privileges moves that are well aligned to the polytope geometry, hence promoting good mixing. The probability with which the sampler selects different bases can be tuned. We present an efficient algorithm for updating the lattice basis, obviating the need for repeated matrix inversion.

Open Problems on Connectivity of Fibers with Positive Margins in Multi-dimensional Contingency Tables

2010 ◽  
Vol 1 (1) ◽  
Author(s):  
Ruriko Yoshida
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Algebraic Theory ◽  
Conditional Inference ◽  
Open Problems ◽  
Markov Basis ◽  
Positive Margins ◽  
Monte Carlo Approach ◽  
The Given

Abstract. Diaconis-Sturmfels developed an algorithm for sampling from conditional distributionsfor a statistical model of discrete exponential families, based on the algebraic theory of toricideals. This algorithm is applied to categorical data analysis through the notion of Markov bases.Initiated with its application to Markov chain Monte Carlo approach for testing statistical fittingof the given model, many researchers have extensively studied the structure of Markov bases formodels in computational algebraic statistics. In the Markov chain Monte Carlo approach fortesting statistical fitting of the given model, a Markov basis is a set of moves connecting allcontingency tables satisfying the given margins. Despite the computational advances, there areapplied problems where one may never be able to compute a Markov basis. In general, the numberof elements in a minimal Markov basis for a model can be exponentially many. Thus, it is importantto compute a reduced number of moves which connect all tables instead of computing a Markovbasis. In some cases, such as logistic regression, positive margins are shown to allow a set ofMarkov connecting moves that are much simpler than the full Markov basis. Such a set is calleda Markov subbasis with assumption of positive margins.In this paper we summarize some computations of and open problems on Markov subbases forcontingency tables with assumption of positive margins under specific models as well as developalgebraic methods for studying connectivity of Markov moves with margin positivity to developMarkov sampling methods for exact conditional inference in statistical models where the Markovbasis is hard to compute.

Edge detection of medical images using Markov basis

2017 ◽  
Vol 11 ◽  
pp. 1825-1833 ◽  
Author(s):  
Husein Hadi Abbass ◽  
Zainab Radhi Mousa
Keyword(s):  
Edge Detection ◽  
Medical Images ◽  
Markov Basis
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