Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algorithm for direct sampling from conditional distributions of toric models

Shuhei Mano, Nobuki Takayama

TL;DR

This work introduces a direct sampling framework for conditional distributions of toric models through contiguity relations of $A$-hypergeometric (GKZ) functions, formulating a Markov chain on a lattice generated by $A$ (the Markov lattice). It links sampling in toric/log-linear models to decomposable graphical models via exact $A$-hypergeometric sum formulae and Pfaffian systems, and provides practical algorithms with complexity insights, plus concrete examples (univariate Poisson regressions, two-way contingency tables, and no-$l$-way interactions). The paper also derives closed-form sum formulas in known cases (e.g., Sundberg’s decomposable-graphical-model formula and Gauss hypergeometric identities) and demonstrates through implementations and benchmarks how direct sampling can outperform Metropolis-based methods in several settings. Overall, it advances exact, structure-exploiting sampling for a broad class of toric models and clarifies the computational bridge between hypergeometric function theory and statistical sampling.

Abstract

We show that contiguity relations of hypergeometric functions of several variables give a direct sampling algorithm from the conditional distribution of toric models in statistics. The algorithm is based on a Markov chain on a lattice generated by a matrix $A$. A correspondence between decomposable graphical models and $A$-hypergeometric systems is discussed. We give a sum formula of special values of $A$-hypergeometric polynomials. Some examples with implementations are presented.

Algorithm for direct sampling from conditional distributions of toric models

TL;DR

This work introduces a direct sampling framework for conditional distributions of toric models through contiguity relations of -hypergeometric (GKZ) functions, formulating a Markov chain on a lattice generated by (the Markov lattice). It links sampling in toric/log-linear models to decomposable graphical models via exact -hypergeometric sum formulae and Pfaffian systems, and provides practical algorithms with complexity insights, plus concrete examples (univariate Poisson regressions, two-way contingency tables, and no--way interactions). The paper also derives closed-form sum formulas in known cases (e.g., Sundberg’s decomposable-graphical-model formula and Gauss hypergeometric identities) and demonstrates through implementations and benchmarks how direct sampling can outperform Metropolis-based methods in several settings. Overall, it advances exact, structure-exploiting sampling for a broad class of toric models and clarifies the computational bridge between hypergeometric function theory and statistical sampling.

Abstract

We show that contiguity relations of hypergeometric functions of several variables give a direct sampling algorithm from the conditional distribution of toric models in statistics. The algorithm is based on a Markov chain on a lattice generated by a matrix . A correspondence between decomposable graphical models and -hypergeometric systems is discussed. We give a sum formula of special values of -hypergeometric polynomials. Some examples with implementations are presented.

Paper Structure

This paper contains 13 sections, 10 theorems, 127 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The direct sampling algorithm
  3. Cases that sum formulae are known
  4. Two-way contingency tables of the independence model
  5. Decomposable graphical models
  6. Examples
  7. Univariate Poisson regressions
  8. Two-way contingency tables
  9. No-$l(\ge 3)$-way interaction models
  10. Discussion
  11. Appendix
  12. Derivation of \ref{['GHTdec']}
  13. The Pfaffian system for the binary no-three-way interaction model

Key Result

Proposition 2.6

When a configuration matrix $A$ is normal, $D/H_A(\beta)$ and $D/H_A(\beta')$ are isomorphic as left $D$ modules for any $\beta, \beta' \in \mathcal{L}_A(b)$. In particular, there exists an isomorphism of the holomorphic solution spaces of them as vector spaces.

Figures (2)

  • Figure 1: The Markov lattice for the matrix $A$ in Example \ref{['exam:2x2:1']} with the maximum $b=(1,2,2,1)^\top$. A sample path is shown by solid edges.
  • Figure 2: The process of removing simplicial vertices from the simplicial complex $[123][124][135][246][247]$.

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.3
  • Remark 2.4
  • Example 2.5: Two-way contingency tables of the independence model
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • Proposition 2.9
  • Definition 3.1
  • ...and 17 more