Removing Phase Transitions from Gibbs Measures

Abstract

Gibbs measures are a fundamental class of distributions for the analysis of high dimensional data. Phase transitions, which are also known as degeneracy in the network science literature, are an emergent property of these models that well describe many physical systems. However, the reach of the Gibbs measure is now far outside the realm of physical systems, and in many of these domains multiphase behavior is a nuisance. This nuisance often makes distribution fitting impossible due to failure of the MCMC sampler, and even when an MLE fit is possible, if the solution is near a phase transition point, the plausibility of the fit can be highly questionable. We introduce a modification to the Gibbs distribution that reduces the effects of phase transitions, and with properly chosen hyper-parameters, provably removes all multiphase behavior. We show that this new distribution is just as easy to fit via MCMCMLE as the Gibbs measure, and provide examples in the Ising model from statistical physics and ERGMs from network science.