Abstract-The partition function of a factor graph and the partition function of the dual factor graph are related to each other by the normal factor graph duality theorem. We apply this result to the classical problem of computing the partition function of the Ising model. In the one-dimensional case, we thus obtain an alternative derivation of the (well-known) analytical solution. In the two-dimensional case, we find that Monte Carlo methods are much more efficient on the dual graph than on the original graph, especially at low temperature.

No abstract

The paper proposes Monte Carlo algorithms for the computation of the information rate of two-dimensional source / channel models. The focus of the paper is on binary-input channels with constraints on the allowed input configurations. The problem of numerically computing the information rate, and even the noiseless capacity, of such channels has so far remained largely unsolved. Both problems can be reduced to computing a Monte Carlo estimate of a partition function. The proposed algorithms use tree-based Gibbs sampling and multilayer (multitemperature) importance sampling. The viability of the proposed algorithms is demonstrated by simulation results.

Tree-based Gibbs sampling (proposed by Hamze and de Freitas) is used to compute a Monte-Carlo estimate of the partition function of factor graphs with cycles. The proposed method can be used, in particular, to compute the capacity of noiseless constrained 2-D channels.