相关论文: Calculating the normalising constant of the Bingha…
We apply the holonomic gradient method to compute the distribution function of a weighted sum of independent noncentral chi-square random variables. It is the distribution function of the squared length of a multivariate normal random…
We propose an accelerated version of the holonomic gradient descent and apply it to calculating the maximum likelihood estimate (MLE) of the Fisher-Bingham distribution on a $d$-dimensional sphere. We derive a Pfaffian system (an integrable…
This paper proposes an efficient numerical integration formula to compute the normalizing constant of Fisher--Bingham distributions. This formula uses a numerical integration formula with the continuous Euler transform to a Fourier-type…
We give a new algorithm to find local maximum and minimum of a holonomic function and apply it for the Fisher-Bingham integral on the sphere $S^n$, which is used in the directional statistics. The method utilizes the theory and algorithms…
This paper is concerned with making Bayesian inference from data that are assumed to be drawn from a Bingham distribution. A barrier to the Bayesian approach is the parameter-dependent normalising constant of the Bingham distribution,…
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…
The Fisher-Bingham distribution ($\mathrm{FB}_8$) is an eight-parameter family of probability density functions (PDF) on $S^2$ that, under certain conditions, reduce to spherical analogues of bivariate normal PDFs. Due to difficulties in…
The Fisher-Bingham system is a system of linear partial differential equations satisfied by the Fisher-Bingham integral for the $n$-dimensional sphere $S^n$. The system is given in [Nakayama et al. (2011), Theorem 2] and it is shown that it…
We apply the holonomic gradient method (HGM) introduced by [9] to the calculation of orthant probabilities of multivariate normal distribution. The holonomic gradient method applied to orthant probabilities is found to be a variant of…
We calculate the integration ideal of annihilating differential operators of the non-normalized Fisher-Bingham distribution and show that the ideal agrees with the set of operators for the Fisher-Bingham integral given in "Holonomic…
We study properties of Fisher distribution (von Mises-Fisher distribution, matrix Langevin distribution) on the rotation group SO(3). In particular we apply the holonomic gradient descent, introduced by Nakayama et al. (2011), and a method…
One of the major problems for maximum likelihood estimation in the well-established directional models is that the normalising constants can be difficult to evaluate. A new general method of "score matching estimation" is presented here on…
We apply the holonomic gradient method introduced by Nakayama et al.(2011) to the evaluation of the exact distribution function of the largest root of a Wishart matrix, which involves a hypergeometric function 1F1 of a matrix argument.…
A new acceptance-rejection method is proposed and investigated for the Bingham distribution on the sphere using the angular central Gaussian distribution as an envelope. It is shown to have high efficiency and to be straightfoward to use.…
We propose a fast algorithm for evaluating the moments of Bingham distribution. The calculation is done by piecewise rational approximation, where interpolation and Gaussian integrals are utilized. Numerical test shows that the algorithm…
Natural gradient methods have been used to optimise the parameters of probability distributions in a variety of settings, often resulting in fast-converging procedures. Unfortunately, for many distributions of interest, computing the…
We present an algorithm for computing a holonomic system for a definite integral of a holonomic function over a domain defined by polynomial inequalities. If the integrand satisfies a holonomic difference-differential system including…
We use the holonomic gradient method to evaluate the probability content of a simplex region under a multivariate normal distribution. This probability equals to the integral of the probability density function of the multivariate Gaussian…
Definite integrals with parameters of holonomic functions satisfy holonomic systems of linear partial differential equations. When we restrict parameters to a one dimensional curve, the system becomes a linear ordinary differential equation…
We give a bijection between a quotient space of the parameters and the space of moments for any $A$-hypergeometric distribution. An algorithmic method to compute the inverse image of the map is proposed utilizing the holonomic gradient…