Supplementary note for “Discrete MRF Inference of Marginal Densities for Non-uniformly Discretized Variable Space”

Detailed derivation of the MF and BP algorithms We show here detailed derivation of the new MF and BP algorithms. In our main paper, we present simplified derivation which needs an assumption that the number S i of mixtures is the same for all sites, i.e., S i = S for any i. In what follows, we present complete derivation of the two algorithms which do not require this assumption; several equations that are omitted in the main paper are also given. Our derivation follows that of conventional MF and BP algorithms in [2]. Derivation of the new MF algorithm As mentioned in our main paper, MF and BP algorithms find P that minimizes the following free energy: F[P] = 〈E〉P − S [P], (30) where the first term is the expectation defined as 〈E〉P = ∫ P(x)E(x)dx, and the second term is the entropy of P, i.e., S [P] = − ∫ P(x) ln P(x)dx. The derivation of MF algorithms start with assuming that the variable of each site i is independent of that of any other site: P(x) ≡ ∏