Holonomic gradient descent and its application to the Fisher–Bingham integral

Nakayama, Hiromasa; Nishiyama, Kenta; Noro, Masayuki; Ohara, Katsuyoshi; Sei, Tomonari; Takayama, Nobuki; Takemura, Akimichi

doi:10.1016/j.aam.2011.03.001

Cited by 73 publications

(108 citation statements)

References 11 publications

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“…The optimal likelihood value rescaled by −n as defined in (16) is 2.457746 = log 11.67846 which is same as the value reported in Nakayama et al (2011). The corresponding quantity for the saddlepoint approximation is 2.463414.…”

Section: Astronomy Datasupporting

confidence: 63%

“…In Nakayama et al (2011), the authors illustrate the general methodology of holonomic gradient method by focusing on two data sets: one from the area of astronomy and the other one from the magnetism. We revisit the first data set in order to confirm that our method gives the same MLE results.…”

Section: Numerical Evidencementioning

confidence: 99%

“…However, recently there has been a renewed interest in this problem with the implementation of the holonomic gradient method (HGM), which in theory is exact since the problem of calculating C is mathematically characterized via a solution of an ODE (see e.g. Nakayama et al 2011;Hashiguchi et al 2013;Sei et al 2013;Koyama 2011;Koyama and Takemura 2016;Koyama et al 2014 andKoyama et al 2012).…”

Section: Introductionmentioning

confidence: 99%

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On the exact maximum likelihood inference of Fisher–Bingham distributions using an adjusted holonomic gradient method

Kume

Sei

2017

Stat Comput

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Holonomic function theory has been successfully implemented in a series of recent papers to efficiently calculate the normalizing constant and perform likelihood estimation for the Fisher-Bingham distributions. A key ingredient for establishing the standard holonomic gradient algorithms is the calculation of the Pfaffian equations. So far, these papers either calculate these symbolically or apply certain methods to simplify this process. Here we show the explicit form of the Pfaffian equations using the expressions from Laplace inversion methods. This improves on the implementation of the holonomic algorithms for these problems and enables their adjustments for the degenerate cases. As a result, an exact and more dimensionally efficient ODE is implemented for likelihood inference.

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Section: Astronomy Datasupporting

confidence: 63%

Section: Numerical Evidencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the exact maximum likelihood inference of Fisher–Bingham distributions using an adjusted holonomic gradient method

Kume

Sei

2017

Stat Comput

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“…There are several applications for Gröbner bases in R. To illustrate, in this section we present an application of approximating a local minimum of a holonomic function g. This is a new method that was introduced in [21] to solve problems in statistics.…”

Section: Gradient Descent For Holonomic Functionsmentioning

confidence: 99%

“…We note that an application of the Fisher-Bingham distribution on a sphere is discussed in [21] and subsequent papers (e.g., [17]). It is no longer of only theoretical interest.…”

Section: Finding a Local Minimum Of A Function Defined By A Definite mentioning

confidence: 99%

Gröbner Basis for Rings of Differential Operators and Applications

Takayama

2013

Gröbner Bases

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We introduce the theory and present some applications of Gröbner bases for the rings of differential operators with rational function coefficients R and for those with polynomial coefficients D.The discussion with R, in the first half, is elementary. In the ring of polynomials, zero-dimensional ideals form the biggest class, and this is also true in R. However, in D, there is no zero-dimensional ideal, and holonomic ideals form the biggest class. Most algorithms for D use holonomic ideals.As an application, we present an algorithm for finding local minimums of holonomic functions; it can be applied to the maximum-likelihood estimate. The last part of this chapter considers A-hypergeometric systems; topics covered in other chapters will reappear in the study of A-hypergeometric systems. We have provided many of the proofs, but some technical proofs in the second half of this chapter have been omitted; these may be found in the references at the end of this chapter.

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Nonlinear Bayesian filtering via holonomic gradient method with quasi moment generating function

Iori

Ohtsuka

2022

Asian Journal of Control

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A symbolic-numeric method is proposed for addressing the Bayesian filtering problems of a class of discrete-time nonlinear stochastic systems. We first approximate the posterior probability density function to be Gaussian. The update law of the mean and variance is formulated as the evaluation of several integrals depending on certain parameters. Unlike existing methods, such as the extended Kalman filter (EKF), unscented Kalman filter (UKF), and particle filter (PF), this formulation considers the nonlinearity of system dynamics exactly. To evaluate the integrals efficiently, we introduce an integral transform motivated by the moment generating function (MGF), which we call a quasi MGF. Furthermore, the quasi MGF is compatible with the Fourier transform of differential operators. We utilize this compatibility to decrease the number of computations of Gröbner bases in the noncommutative rings of differential operators, which reduces the offline computational time. A numerical example is presented to show the efficiency of the proposed method compared to that of other existing methods such as the EKF, UKF, and PF.

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Holonomic gradient descent and its application to the Fisher–Bingham integral

Cited by 73 publications

References 11 publications

On the exact maximum likelihood inference of Fisher–Bingham distributions using an adjusted holonomic gradient method

On the exact maximum likelihood inference of Fisher–Bingham distributions using an adjusted holonomic gradient method

Gröbner Basis for Rings of Differential Operators and Applications

Nonlinear Bayesian filtering via holonomic gradient method with quasi moment generating function

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