Granger causality and the inverse Ising problem

The inference of the couplings of an Ising model with given means and correlations is called inverse Ising problem. This approach has received a lot of attention as a tool to analyze neural data. We show that autoregressive methods may be used to learn the couplings of an Ising model, also in the case of asymmetric connections and for multi-spin interactions. We find that, for each link, the linear Granger causality is two times the corresponding transfer entropy (i.e. the information flow on that link) in the weak coupling limit. For sparse connections and a low number of samples, the l1 regularized least squares method is used to detect the interacting pairs of spins. Nonlinear Granger causality is related to multispin interactions.