Monte Carlo Methods for the Ferromagnetic Potts Model Using Factor Graph Duality

Abstract

Normal factor graph duality offers new possibilities for Monte Carlo algorithms in graphical models. Specifically, we consider the problem of estimating the partition function of the ferromagnetic Ising and Potts models by Monte Carlo methods, which are known to work well at high temperatures but to fail at low temperatures. We propose Monte Carlo methods (uniform sampling and importance sampling) in the dual normal factor graph and demonstrate that they behave differently: they work particularly well at low temperatures. By comparing the relative error in estimating the partition function, we show that the proposed importance sampling algorithm significantly outperforms the state-of-the-art deterministic and Monte Carlo methods. For the ferromagnetic Ising model in an external field, we show the equivalence between the valid configurations in the dual normal factor graph and the terms that appear in the high-temperature series expansion of the partition function. Following this result, we discuss connections with Jerrum–Sinclair’s polynomial randomized approximation scheme (the subgraphs-world process) for evaluating the partition function of ferromagnetic Ising models.