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Recursions on the marginals and exact computation of the normalizing constant for Gibbs processes

Cécile Hardouin, Xavier Guyon

TL;DR

The paper tackles the exact computation of marginals and the normalizing constant $C$ for Gibbs distributions by exploiting Markov properties through a matrix formulation. It develops forward recursions and future-conditioning recursions to compute marginals on temporal sequences and general subsets, and extends these methods to $r$-range potentials and to spatial Gibbs fields treated as multidimensional processes. The main contributions are a compact matrix-product expression for $C$, a future-conditioned marginal recursion, a dichotomous marginals sequence, and Ising-model demonstrations with guidance on when to power versus diagonalize, plus generalizations to higher-order potentials. These methods offer exact computation capabilities that can be accelerated by low-rank and randomized linear algebra, with applicability to large lattices.

Abstract

This paper presents different recursive formulas for computing the marginals and the normalizing constant of a Gibbs distribution $π$: The common thread is the use of the underlying Markov properties of such processes. The procedures are illustrated with several examples, particularly the Ising model.

Recursions on the marginals and exact computation of the normalizing constant for Gibbs processes

TL;DR

The paper tackles the exact computation of marginals and the normalizing constant for Gibbs distributions by exploiting Markov properties through a matrix formulation. It develops forward recursions and future-conditioning recursions to compute marginals on temporal sequences and general subsets, and extends these methods to -range potentials and to spatial Gibbs fields treated as multidimensional processes. The main contributions are a compact matrix-product expression for , a future-conditioned marginal recursion, a dichotomous marginals sequence, and Ising-model demonstrations with guidance on when to power versus diagonalize, plus generalizations to higher-order potentials. These methods offer exact computation capabilities that can be accelerated by low-rank and randomized linear algebra, with applicability to large lattices.

Abstract

This paper presents different recursive formulas for computing the marginals and the normalizing constant of a Gibbs distribution : The common thread is the use of the underlying Markov properties of such processes. The procedures are illustrated with several examples, particularly the Ising model.

Paper Structure

This paper contains 14 sections, 3 theorems, 33 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Recursions for a temporal Gibbs distribution
  3. Factoring joint distribution, Markov field and Markov chain properties
  4. Forward recursions over marginal distributions
  5. Examples and numerical performances
  6. Results for $r$-range potentials
  7. Other recursions
  8. A new recursive algorithm for marginals based on future conditionals
  9. Dichotomous marginals sequence
  10. The case of spatial Gibbs fields
  11. A temporal multidimensional Gibbs process
  12. Computing the normalization constant for an Ising model
  13. Some generalizations
  14. Conclusion

Key Result

proposition thmcounterproposition

1- The normalizing constant of the general model (gibbs) is given by 2- For time invariant potentials , the formula reduces to $C=\mathbf{1}^{\mathrm{T}}(H)^{T-1}\mathbf{1.}$

Theorems & Definitions (3)

  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proposition thmcounterproposition