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We prove that the marginal densities of a global probability mass function ina primal normal factor graph and the corresponding marginal densities in thedual normal factor graph are related via local mappings. The mapping depends onthe Fourier transform of the local factors of the models. Details of themapping, including its fixed points, are derived for the Ising model, and thenextended to the Potts model. By employing the mapping, we can transformsimultaneously all the estimated marginal densities from one domain to theother, which is advantageous if estimating the marginals can be carried outmore efficiently in the dual domain. An example of particular significance isthe ferromagnetic Ising model in a positive external field, for which there isa rapidly mixing Markov chain (called the subgraphs-world process) to generateconfigurations in the dual normal factor graph of the model. Our numericalexperiments illustrate that the proposed procedure can provide more accurateestimates of marginal densities in various settings.

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