Title: Inverse Ising problem in continuous time: A latent variable approach,
Abstract: We consider the inverse Ising problem, i.e. the inference of network
couplings from observed spin trajectories for a model with continuous time
Glauber dynamics. By introducing two sets of auxiliary latent random variables
we render the likelihood into a form, which allows for simple iterative
inference algorithms with analytical updates. The variables are: (1) Poisson
variables to linearise an exponential term which is typical for point process
likelihoods and (2) Pólya-Gamma variables, which make the likelihood
quadratic in the coupling parameters. Using the augmented likelihood, we derive
an expectation-maximization (EM) algorithm to obtain the maximum likelihood
estimate of network parameters. Using a third set of latent variables we extend
the EM algorithm to sparse couplings via L1 regularization. Finally, we develop
an efficient approximate Bayesian inference algorithm using a variational
approach. We demonstrate the performance of our algorithms on data simulated
from an Ising model. For data which are simulated from a more biologically
plausible network with spiking neurons, we show that the Ising model captures
well the low order statistics of the data and how the Ising couplings are
related to the underlying synaptic structure of the simulated network.