Marginal Densities, Factor Graph Duality, and High-Temperature Series Expansions

TL;DR

This paper establishes a local mapping between marginal densities in primal and dual normal factor graphs, enabling more efficient and accurate estimation of marginals, especially for models like the Ising and Potts models.

Contribution

It introduces a novel local mapping based on Fourier transforms that relates marginals in primal and dual graphs, improving estimation efficiency.

Findings

The mapping depends on the Fourier transform of local factors.

Numerical experiments show improved marginal density estimates.

Application to Ising and Potts models demonstrates effectiveness.

Abstract

We prove that the marginal densities of a global probability mass function in a primal normal factor graph and the corresponding marginal densities in the dual normal factor graph are related via local mappings. The mapping depends on the Fourier transform of the local factors of the models. Details of the mapping, including its fixed points, are derived for the Ising model, and then extended to the Potts model. By employing the mapping, we can transform simultaneously all the estimated marginal densities from one domain to the other, which is advantageous if estimating the marginals can be carried out more efficiently in the dual domain. An example of particular significance is the ferromagnetic Ising model in a positive external field, for which there is a rapidly mixing Markov chain (called the subgraphs-world process) to generate configurations in the dual normal factor graph of the model. Our numerical experiments illustrate that the proposed procedure can provide more accurate estimates of marginal densities in various settings.