Approximations in the Homogeneous Ising Model with Application to Scene Analysis

The Ising model is important in statistical modeling and inference in many applications, however, its normalizing constant, mean number of active vertices and mean spin interaction—quantities often needed in inference—are computationally intractable. We provide accurate approximations that make it possible to numerically calculate these quantities in the homogeneous case. Simulation studies indicate good performance of our approximation formulae that are scalable and unfazed by the size (number of nodes, degree of graph) of the Markov Random Field. The practical import of our approximation formulae is illustrated in performing Bayesian inference in a functional Magnetic Resonance Imaging activation detection experiment, in likelihood ratio testing, for anisotropy in the spatial patterns of yearly increases in pistachio tree yields, and for independence of the least significant bit in the three color channels of a gigapixel image. Supplementary materials for this article are available online.