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We consider the setting where a collection of time series, modeled as randomprocesses, evolve in a causal manner, and one is interested in learning thegraph governing the relationships of these processes. A special case of wideinterest and applicability is the setting where the noise is Gaussian andrelationships are Markov and linear. We study this setting with two additionalfeatures: firstly, each random process has a hidden (latent) state, which weuse to model the internal memory possessed by the variables (similar to hiddenMarkov models). Secondly, each variable can depend on its latent memory statethrough a random lag (rather than a fixed lag), thus modeling memory recallwith differing lags at distinct times. Under this setting, we develop anestimator and prove that under a genericity assumption, the parameters of themodel can be learned consistently. We also propose a practical adaption of thisestimator, which demonstrates significant performance gains in both syntheticand real-world datasets.

Learning Temporal Dependence from Time-Series Data with Latent Variables