A class of random fields on complete graphs with tractable partition function

TL;DR

This paper identifies a class of random fields on complete and bipartite graphs where the partition function and marginals can be computed efficiently, aiding in evaluating approximation algorithms.

Contribution

It introduces a polynomial-time method for computing partition functions and marginals for specific classes of random fields on complete and bipartite graphs.

Findings

Partition function and marginals computed in polynomial time for certain random fields.

Includes Ising models with homogeneous pairwise and arbitrary unary potentials.

Facilitates exact error estimation for approximation algorithms.

Abstract

The aim of this short note is to draw attention to a method by which the partition function and marginal probabilities for a certain class of random fields on complete graphs can be computed in polynomial time. This class includes Ising models with homogeneous pairwise potentials but arbitrary (inhomogeneous) unary potentials. Similarly, the partition function and marginal probabilities can be computed in polynomial time for random fields on complete bipartite graphs, provided they have homogeneous pairwise potentials. We expect that these tractable classes of large scale random fields can be very useful for the evaluation of approximation algorithms by providing exact error estimates.