Gaussian time-series models are often specified through their spectral density. Such models pose several computational challenges, in particular because of the non-sparse nature of the covariance matrix. We derive a fast approximation of the likelihood for such models. We use importance sampling to correct for the approximation error. We show that the variance of the importance sampling weights vanishes as the sample size goes to infinity. We show that the posterior is typically multi-modal, and derive a Sequential Monte Carlo sampler based on an annealing sequence in order to sample from the approximate posterior. Performance of the overall approach is evaluated on simulated and real datasets.
