Holonomic modules associated with multivariate normal probabilities of polyhedra

@article{Koyama2013HolonomicMA,
  title={Holonomic modules associated with multivariate normal probabilities of polyhedra},
  author={Tamio Koyama},
  journal={arXiv: Classical Analysis and ODEs},
  year={2013}
}
  • Tamio Koyama
  • Published 27 November 2013
  • Mathematics
  • arXiv: Classical Analysis and ODEs
The probability content of a convex polyhedron with a multivariate normal distribution can be regarded as a real analytic function. We give a system of linear partial differential equations with polynomial coefficients for the function and show that the system induces a holonomic module. The rank of the holonomic module is equal to the number of nonempty faces of the convex polyhedron, and we provide an explicit Pfaffian equation (an integrable connection) that is associated with the holonomic… 
View PDF on arXiv
2 Citations
Background Citations
1
Results Citations
1

Figures from this paper

2 Citations

Holonomic gradient method for the probability content of a simplex region with a multivariate normal distribution

We use the holonomic gradient method to evaluate the probability content of a simplex region with a multivariate normal distribution. This probability equals to the integral of the probability…

Algebraic statistics

This tutorial will consist of a detailed study of two examples where the algebra/statistics connection has proven especially useful: in the study of phylogenetic models and in the analysis of contingency tables.

26 References

Pfaffian Systems of A-Hypergeometric Equations

These theoretical study gives algorithms of transforming a given A-hypergeometric system into a Pfaffian system, which has a combinatorial description of hypergeometric systems associated to a class of order polytopes.

The holonomic gradient method for the distribution function of the largest root of a Wishart matrix

Computation of the characteristic variety and the singular locus of a system of differential equations with polynomial coefficients

  • T. Oaku
  • Mathematics, Computer Science
  • 1994
It is proved that for a system of linear partial differential equations with polynomial coefficients, the Gröbner basis in the Weyl algebra is sufficient for the computation of the characteristic…

Abstract tubes associated with perturbed polyhedra with applications to multidimensional normal probability computations

Let $K$ be a closed convex polyhedron defined by a finite number of linear inequalities. In this paper we refine the theory of abstract tubes (Naiman and Wynn, 1997) associated with $K$ when $K$ is…

Calculation of orthant probabilities by the holonomic gradient method

The results show that multivariate normal orthant probabilities possess some remarkable properties from the viewpoint of holonomic systems and the numerical performance of the holonomic gradient method is comparable or superior to that of existing methods.

Configurations and Invariant Gauss-Manin Connections of Integrals I

satisfy Gauss-Manin connections or equivalently holonomic systems in the sense of S. S. K. (See [8], [16] and [18].) But generally it seems difficult to get their explicit formulae in global forms.…

Algorithms forb-Functions, Restrictions, and Algebraic Local Cohomology Groups ofD-Modules

Our purpose is to present algorithms for computing some invariants and functors attached to algebraic D-modules by using Grobner bases for differential operators. Let K be an algebraically closed…

ALGEBRAIC ALGORITHMS FOR D-MODULES AND NUMERICAL ANALYSIS

This paper shows that these algorithms give correct algorithms to perform several operations for holonomic functions and also generates substantial information for numerical evaluation ofholonomic functions.

The union of balls and its dual shape

Efficient algorithms are described for computing topological, combinatorial, and metric properties of the union of finitely many spherical balls in ℝd. These algorithms are based on a simplicial…

Abstract tubes, improved inclusion-exclusion identities and inequalities and importance sampling

Numerous statistical applications require the evaluation of the probability content of a convex polyhedron. We demonstrate for a given polyhedron in R d that there is a depth d inclusion-exclusion…