Erik Aurell: Recent developments on the inverse Ising problem

The probability distributions over a set of binary variables which have given single-variable marginals and given pair-wise correlations, and which maximize entropy, from an exponential family called the Ising model of statistical mechanics. The (standard) Ising problem is to compute properties (marginals, correlation functions) in such a model at given model parameters. The inverse Ising problem is the opposite task to infer parameter given marginals and correlation functions. In this exponential family (sample) marginals and (sample) pair-wise correlation functions are sufficient statistics, but a maximum likelihood solution is not feasible in large instances since the normalization factor (the partition function) is not easily computable.

Various approximations which allow for fast (but not exact) solutions have been introduced in the last decade. The competitive evaluations of the various scheme is at this point mainly empirical, by numerical experiments in well-controlled test examples, or in concrete applications, but some rigorous results have also been obtained. I will survey these developments, with an emphasis on what my group has been working on.

This is joint work with John Hertz, Yasser Roudi, Mikko Alava, Zeng HongLi and others.