In this paper we consider the problem of inferring a causality structure from multiple time series with asynchronous sampling by use of the Kinetic Ising Model. The problem is ubiquitous in many fields of science, ranging from neuroscience ― where spike train data from multiple neurons can carry information about connections in the brain ― to finance, where one is interested in forecasting the behavior of observable variables (such as prices and order flows) and often has to take into account multiple time varying factors. A recent stream of literature has started considering the problem when some variables are not observable. We expand from there and consider a case where observations are asynchronous or incomplete, thus at each time frame one has a partial snapshot of the whole system. By adopting an Expectation-Maximization-like algorithm for the log-likelihood maximization and deriving the TAP equations, we obtain impressive results both in terms of reconstruction of a weighted network and of inference of hidden variable values even when small fractions of the configurations are observed in each frame. We present results for synthetic data, exploring different regimes of sparsity in the underlying network and different specifications of missing data. We aim to apply the methodology in the immediate future to financial data of trader activity and provide a description of how information propagates between agents in the market.
