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

We prove that the marginals densities of a primal normal factor graph and the corresponding marginal densities of its dual normal factor graph are related via local mappings. The mapping relies on no assumptions on the size, on the topology, or on the parameters of the graphical model. The mapping provides us with a simple procedure to transform simultaneously the estimated marginals from one domain to the other, which is particularly useful when such computations can be carried out more efficiently in one of the domains. In the case of the Ising model, valid configurations in the dual normal factor graph of the model coincide with the terms that appear in the high-temperature series expansion of the partition function. The subgraphs-world process (as a rapidly mixing Markov chain) can therefore be employed to draw samples according to the global probability mass function of the dual normal factor graph of ferromagnetic Ising models.