  We propose a new algorithm for inferring the state of hidden spins and
reconstructing the connections in a synchronous kinetic Ising model, given the
observed history. Focusing on the case in which the hidden spins are
conditionally independent of each other given the state of observable spins, we
show that calculating the likelihood of the data can be simplified by
introducing a set of replicated auxiliary spins. Belief Propagation (BP) and
Susceptibility Propagation (SusP) can then be used to infer the states of
hidden variables and learn the couplings. We study the convergence and
performance of this algorithm for networks with both Gaussian-distributed and
binary bonds. We also study how the algorithm behaves as the fraction of hidden
nodes and the amount of data are changed, showing that it outperforms the TAP
equations for reconstructing the connections.
