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We consider the inverse Ising problem, i.e. the inference of networkcouplings from observed spin trajectories for a model with continuous timeGlauber dynamics. By introducing two sets of auxiliary latent random variableswe render the likelihood into a form, which allows for simple iterativeinference algorithms with analytical updates. The variables are: (1) Poissonvariables to linearise an exponential term which is typical for point processlikelihoods and (2) P\'olya-Gamma variables, which make the likelihoodquadratic in the coupling parameters. Using the augmented likelihood, we derivean expectation-maximization (EM) algorithm to obtain the maximum likelihoodestimate of network parameters. Using a third set of latent variables we extendthe EM algorithm to sparse couplings via L1 regularization. Finally, we developan efficient approximate Bayesian inference algorithm using a variationalapproach. We demonstrate the performance of our algorithms on data simulatedfrom an Ising model. For data which are simulated from a more biologicallyplausible network with spiking neurons, we show that the Ising model captureswell the low order statistics of the data and how the Ising couplings arerelated to the underlying synaptic structure of the simulated network.

Inverse Ising problem in continuous time: A latent variable approach