Likelihood Analysis of Power Spectra and Generalized Moment Problems

We develop an approach to the spectral estimation that has been advocated by [A. Ferrante et al., “Time and spectral domain relative entropy: A new approach to multivariate spectral estimation,” IEEE Trans. Autom. Control, vol. 57, no. 10, pp. 2561-2575, Oct. 2012.] and, in the context of the scalar-valued covariance extension problem, by [P. Enqvist and J. Karlsson, “Minimal itakura-saito distance and covariance interpolation,” in Proc. 47th IEEE Conf. Decision Control, 2008, pp. 137-142]. The aim is to determine the power spectrum that is consistent with given moments and minimizes the relative entropy between the probability law of the underlying Gaussian stochastic process to that of a prior. The approach is analogous to the framework of earlier work by Byrnes, Georgiou, and Lindquist and can also be viewed as a generalization of the classical work by Burg and Jaynes on the maximum entropy method. In this paper, we present a new fast algorithm in the general case (i.e., for general Gaussian priors) and show that for priors with a specific structure the solution can be given in closed form.