Learning and Testing Inverse Statistical Problems For Interacting Systems Undergoing Phase Transition

TL;DR

This paper explores inverse problems in statistical mechanics, applying maximum likelihood, pseudo-likelihood, and mean-field methods to infer parameters in models undergoing phase transitions, with practical demonstrations on various systems.

Contribution

It provides a comprehensive theoretical and practical comparison of inference methods applied to phase transition models, including implementation resources.

Findings

Effective inference of model parameters demonstrated

Comparison of methods highlights strengths and limitations

Open-source tools facilitate further experimentation

Abstract

Inverse problems arise in situations where data is available, but the underlying model is not. It can therefore be necessary to infer the parameters of the latter starting from the former. Statistical mechanics offers a toolbox of techniques to address this challenge. In this work, we illustrate three of the main methods: the Maximum Likelihood, Maximum Pseudo-Likelihood, and Mean-Field approaches. We begin with a thorough theoretical introduction to these methods, followed by their application to inference in several well-known statistical physics systems undergoing phase transitions. Namely, we consider the ordered and disordered Ising models, the vector Potts model, and the Blume-Capel model on both regular lattices and random graphs. This discussion is accompanied by a GitHub repository that allows users to both reproduce the results and experiment with new systems.