Pfaffian Systems of A-Hypergeometric Systems II --- Holonomic Gradient Method

Ohara, Katsuyoshi; Takayama, Nobuki

doi:10.48550/arxiv.1505.02947

Cited by 6 publications

(10 citation statements)

References 10 publications

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“…and E r is the r × r identity matrix. Ohara and Takayama [20] showed that q 0 (b; y) = Z A (b; y) is obtained by the matrix multiplications:…”

Section: Algorithm 21 ([15])mentioning

confidence: 99%

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Algebraic algorithm for direct sampling from toric models

Mano¹,

Takayama²

2021

Preprint

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We show that Pfaffians or contiguity relations of hypergeometric functions of several variables give a direct sampling algorithm from toric models in statistics, which is a Markov chain on a lattice generated by a matrix A. A correspondence among graphical toric models and A-hypergeometric system is discussed and we give a sum formula of special values of A-hypergeometric polynomials. Some hypergeometric series which are interesting in view of statistics are presented.

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“…and E r is the r × r identity matrix. Ohara and Takayama [20] showed that q 0 (b; y) = Z A (b; y) is obtained by the matrix multiplications:…”

Section: Algorithm 21 ([15])mentioning

confidence: 99%

“…In either case, r 2 n dominates for large r, because r grows rapidly than r 1 r 2 . For example, (r 1 r 2 , r) = (4, 2), (9,6), (16,20), (25,70) for r 1 = r 2 = 2, 3, 4, 5, respectively. The unique minimal Markov basis up to sign is B = (z ij : i ∈ [r 1 ], j ∈ [r 2 ]), where…”

Section: Two-way Contingency Tablesmentioning

confidence: 99%

Algebraic algorithm for direct sampling from toric models

Mano¹,

Takayama²

2021

Preprint

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“…There are several algorithms to obtain contiguity relations [32], [24], [23], [7]. Among them, we choose to use the method of twisted cohomology groups given in [7], because it is the most efficient method for the case of two-way contingency tables.…”

Section: Contiguity Relationmentioning

confidence: 99%

Holonomic Gradient Method for Two Way Contingency Tables

Tachibana,

Goto,

Koyama

et al. 2018

Preprint

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The holonomic gradient method gives an algorithm to efficiently and accurately evaluate normalizing constants and their derivatives. We apply the holonomic gradient method in the case of the conditional Poisson or multinomial distribution on two way contingency tables. We utilize the modular method in computer algebra or some other tricks for an efficient and exact evaluation, and we compare them and discuss on their implementation. We also discuss on a theoretical aspect of the distribution from the viewpoint of the conditional maximum likelihood estimation. We decompose parameters of interest and nuisance parameters in terms of sigma algebras for general two way contingency tables with arbitrary zero cell patterns.

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“…To compute the normalizing constant of the two-way contingency tables with fixed marginal sums, another type of HGM algorithm, which is based on difference equations among A-hypergeometric polynomials, was employed [46]. Following [12], we call this method the difference HGM.…”

Section: Computation Of A-hypergeometric Polynomialsmentioning

confidence: 99%

“…By virtue of this explicit expression, alternative algebraic methods for evaluating the A-hypergeometric polynomials are presented. They are examples of methods called the holonomic gradient methods (HGMs) [11,12,13]. Roughly speaking, the difference HGM demands less computational cost, while the recurrence relation gives more accurate estimates.…”

Section: Introductionmentioning

confidence: 99%

Partition structure and the $A$-hypergeometric distribution associated with the rational normal curve

Mano¹

2017

Electron. J. Statist.

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A distribution whose normalization constant is an A-hypergeometric polynomial is called an A-hypergeometric distribution. Such a distribution is in turn a generalization of the generalized hypergeometric distribution on the contingency tables with fixed marginal sums. In this paper, we will see that an A-hypergeometric distribution with a homogeneous matrix of two rows, especially, that associated with the rational normal curve, appears in inferences involving exchangeable partition structures. An exact sampling algorithm is presented for the general (any number of rows) A-hypergeometric distributions. Then, the maximum likelihood estimation of the A-hypergeometric distribution associated with the rational normal curve, which is an algebraic exponential family, is discussed. The information geometry of the Newton polytope is useful for analyzing the full and the curved exponential family. Algebraic methods are provided for evaluating the A-hypergeometric polynomials. MSC2010: 62E15,13P25,60C05

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Pfaffian Systems of A-Hypergeometric Systems II --- Holonomic Gradient Method

Cited by 6 publications

References 10 publications

Algebraic algorithm for direct sampling from toric models

Algebraic algorithm for direct sampling from toric models

Holonomic Gradient Method for Two Way Contingency Tables

Partition structure and the $A$-hypergeometric distribution associated with the rational normal curve

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