Bayesian Structure Learning for Stationary Time Series

While much work has explored probabilistic graphical models for independent data, less attention has been paid to time series. The goal in this setting is to determine conditional independence relations between entire time series, which for stationary series, are encoded by zeros in the inverse spectral density matrix. We take a Bayesian approach to structure learning, placing priors on (i) the graph structure and (ii) spectral matrices given the graph. We leverage a Whittle likelihood approximation and define a conjugate prior---the hyper complex inverse Wishart---on the complex-valued and graph-constrained spectral matrices. Due to conjugacy, we can analytically marginalize the spectral matrices and obtain a closed-form marginal likelihood of the time series given a graph. Importantly, our analytic marginal likelihood allows us to avoid inference of the complex spectral matrices themselves and places us back into the framework of standard (Bayesian) structure learning. In particular, combining this marginal likelihood with our graph prior leads to efficient inference of the time series graph itself, which we base on a stochastic search procedure, though any standard approach can be straightforwardly modified to our time series case. We demonstrate our methods on analyzing stock data and neuroimaging data of brain activity during various auditory tasks.