Inverse Ising problem in continuous time: A latent variable approach

In recent years, the inverse Ising problem, i.e. the reconstruction of couplings and external fields of an Ising model from samples of spin configurations, has attracted considerable interest in the physics community [1]. This is due to the fact that Ising models play an important role for data modeling with applications to neural spike data [2, 3], protein structure determination [4], and gene expression analysis [5]. Much effort has been devoted to the development of algorithms for the static inverse Ising problem. This is a nontrivial task, because statistically efficient, likelihood based methods become computationally infeasible by the intractability of the partition function of the model. Hence one has to resort to either approximate inference methods or to other statistical estimators such as pseudo-likelihood methods [6], or the interaction screening algorithm [7]. The situation is somewhat simpler for the dynamical inverse Ising problem, which recently attracted attention [8-13]. If one assumes a Markovian dynamics, the exact normalisation of the spin transition probabilities allows for an explicit computation of the likelihood if one has a complete set of observed data over time. Nevertheless, the model parameters enter the likelihood in a fairly complex way, and the application of more advanced statistical approaches such as Bayesian inference again becomes a nontrivial task. This is especially true for the continuous time kinetic Ising model where the spins are governed by Glauber dynamics [14].