Direct sampler

Direct sampler by contiguity relations of GKZ hypergeometric systems

Version 0.1

November 3, 2025

Copyright © Risa/Asir committers 2004–2025. All rights reserved.

1 About this document

This document explains Risa/Asir functions for direct sampler by contiguity relations of GKZ hypergeometric systems
Loading the package:

import("gtt_ds.rr");

References cited in this document.

• [MT2021v2] S.Mano, N.Takayama, Algorithm for direct sampling from conditional distributions of toric models, version 2, https://arxiv.org/abs/2110.14992.

2 Simplicial complex to A matrix

2.1 gtt_ds.getA

gtt_ds.getA(Levels,Facets)

:: It returns a matrix A standings for a simplicial complex defined by Facets and the set of levels of each vertex Levels.

return

Matrix A to define a GKZ system.

Levels

Levels of each vertex of the simplicial complex Facets.

Facets

The set of facets of a simplicial complex. A facet is expressed by a 0, 1 vector. 1 stands for the vertex of the position belongs to the facet and 0 stands for the vertex of the position does not belong to the facet.

• The following is an explanation of how facets are expressed by the varible Facets. Consider the simplicial complex of 4 vertices [123][134] (square with one diagonal line)

   1      2
   *------*
   |\     |
   | \    |
   |  \   |
   |   \  |
   |    \ |
   *------*
   4      3
  
  

  The facet 12 is expressed as [1,1,0,0], The facet 23 is expressed as [0,1,1,0], the facet 13 is expressed as [1,0,1,0]. The set of all facets is [[1,1,0,0],[0,1,1,0],[1,0,1,0],[1,0,0,1],[0,0,1,1]].

• The variable Levels is a set of levels of each vertex by pressed by numbers 1, 2, 3, ... For example, if we have four vertices and Levels is [2,2,3,4], then the vertex 1 has the levels 1,2, the vertex 2 also has the levels 1,2, the vertex 3 has the levels 1,2,3, and the vertex 4 ahs the levels 1,2,3,4.
• The rows of the matrix A is indexed by all levels of all facets. See page 4 of [MT2021v2], {\cal I}_F . Let us show an example by our four vertices simplicial complex above. Let F be the facet [13] and Levels [2,2,3,4]. Then, all levels of the facet F is [1,0,1,0],[1,0,2,0],[1,0,3,0],[2,0,1,0],[2,0,2,0],[2,0,3,0]. The rows of the matrix A is indexed by such index that runs over all facets.
• For a graphical model, the set of facets are the set of cliques.
• The columns of the matrix A is indexed by all possible combinations of levels. See page 4 of [MT2021v2], {cal I}_V. For example, if Levels=[2,4], then the column index is [1,1],[1,2],[1,3],[1,4],[2,1],[2,2],[2,3],[2,4].
• The matrix A is a 0-1 matrix. If the row index and the column index matches on the facet F underlying the row index, the the entry of A is 1.

Example: Consider the simplicial complex [1][2] (independent two points) with Levels=[2,2]. Then, the rows of the matrix A is indexed by [1,0],[2,0],[0,1],[0,2] and the columns of the matrix A is indexed by [1,1],[1,2],[2,1],[2,2]. The matrix A is \pmatrix{1&1&0&0\cr 0&0&1&1\cr 1&0&1&0\cr 0&1&0&1\cr } For example, [1,0],[1,1] matches (1?,11, replace 0 by the wild card ? and regard it a regular expression), [1,0],[1,2] also matches (1?,12), but [1,0],[0,1] does not match (1?,01).

[2122] import("gtt_ds.rr");
[4432] gtt_ds.getA([2,2],[[1,0],[0,1]]);
[ 1 1 0 0 ]
[ 0 0 1 1 ]
[ 1 0 1 0 ]
[ 0 1 0 1 ]

Example: Consider the simplicial complex [1][2] (independent two points) with Levels=[2,3]. Then, the rows of the matrix A is indexed by [1,0],[2,0],[0,1],[0,2],[0,3] and the columns of the matrix A is indexed by [1,1],[1,2],[1,3],[2,1],[2,2],[2,3]. The matrix A is \pmatrix{ 1& 1& 1& 0& 0& 0 \cr 0& 0& 0& 1& 1& 1 \cr 1& 0& 0& 1& 0& 0 \cr 0& 1& 0& 0& 1& 0 \cr 0& 0& 1& 0& 0& 1 \cr }

[2122] import("gtt_ds.rr");
[4432]  gtt_ds.getA([2,3],[[1,0],[0,1]]);
[ 1 1 1 0 0 0 ]
[ 0 0 0 1 1 1 ]
[ 1 0 0 1 0 0 ]
[ 0 1 0 0 1 0 ]
[ 0 0 1 0 0 1 ]

Example: Consider the simplicial complex [123][134] with Levels=[2,2,3,4]. For example, the row index [1,2,2,0] and the column index [1,1,1,1]. They do not match (122? and 1111). The row index [1,2,2,0] and the column index [1,2,2,4]. matches (122? and 1114).

The row index [1,1,1,0] and the column index [1,1,1,1]. matches (111? and 1111). The row index [1,1,1,0] and the column index [1,1,1,2]. matches (111? and 1112). ... The row index [1,1,1,0] and the column index [1,2,1,1]. does not matche (111? and 1211). ... The row index [1,1,1,0] and the column index [2,2,3,4]. does not matche (111? and 2234).

[2122] import("gtt_ds.rr");
[4432] gtt_ds.getA([2,2,3,4],[[1,1,1,0],[1,0,1,1]]);
[ 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
                         0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]
[ 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
                         0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]
--- snip ---
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
                         0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 ]

Example: See pages 20, 21 of [MT2021v2].

[2123] import("mt_mm.rr");
[4432] gtt_ds.getA([2,2,2,2],[[1,1,1,0],[1,1,0,1]]);
[ 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]
[ 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 ]
[ 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ]
[ 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 ]
[ 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 ]
[ 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 ]
[ 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 ]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 ]
[ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ]
[ 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ]
[ 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 ]
[ 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 ]
[ 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 ]
[ 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 ]
[ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 ]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 ]

Refer to

gtt_ds.direct_sampler

3 Direct sampler of two way contingency table

3.1 gtt_ds.direct_sampler

gtt_ds.direct_sampler(MarginalSum,CellProb)

:: It returns a sample contingency table.

return

A sample contingency table.

MarginalSum

Marginal sums of a two way contingency table.

CellProb

A list of probabilities of each cell of the table.

• Repeating to call this function generates samples with integral entries with given marginal sums MarginalSum.
• The function utilizes the package gtt_ekn3.rr to obtain contiguity relations for the GKZ hypergeometric systems (Aomoto-Gel’fand system) for two way contingency tables.

Example: we consider 2 \times 3 contingency tables with the cell probability \pmatrix{ 1& 1/2 & 1/3 \cr 1& 1 & 1 \cr } , the row marginal sums (5,6), and the column marignal sums (3,3,5).

[1883] import("gtt_ds.rr");
[4447] for (I=0; I<3; I++) 
  printf("%a\n-----\n",gtt_ds.direct_sampler(MarginalSum,CellProb));  
[ 2 0 3 ]
[ 1 3 2 ]
-----
[ 2 2 1 ]
[ 1 1 4 ]
-----
[ 2 0 3 ]
[ 1 3 2 ]
-----

Todo, install a function to obtain contiguity relations for a given matrix A. Call this function with optional variable a (A matrix), MarginalSum is \beta. Refer to mt_mm.rr and latest isom-project codes.

Refer to

gtt_ds.getA

Index

Table of Contents

Short Table of Contents

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