Precision spectral estimation at sub-Hz frequencies: closed-form posteriors and Bayesian noise projection

We present a Bayesian method for estimating spectral quantities in multivariate Gaussian time series. The approach, based on periodograms and Wishart statistics, yields closed-form expressions at any given frequency for the marginal posterior distributions of the individual power spectral densities, the pairwise coherence, and the multiple coherence, as well as for the joint posterior distribution of the full cross-spectral density matrix. In the context of noise projection - where one series is modeled as a linear combination of filtered versions of the others, plus a background component - the method also provides closed-form posteriors for both the susceptibilities, i.e., the filter transfer functions, and the power spectral density of the background. Originally developed for the analysis of the data from the European Space Agency's LISA Pathfinder mission, the method is particularly well-suited to very-low-frequency data, where long observation times preclude averaging over large sets of periodograms, which would otherwise allow these to be treated as approximately normally distributed.