Bayesian nonparametric spectral density estimation using B-spline priors

We present a new Bayesian nonparametric approach to estimating the spectral density of a stationary time series. A nonparametric prior based on a mixture of B-spline distributions is specified and can be regarded as a generalization of the Bernstein polynomial prior of Petrone (1999a,b) and Choudhuri et al. (2004). Whittle's likelihood approximation is used to obtain the pseudo-posterior distribution. This method allows for a data-driven choice of the smoothing parameter as well as the number and the location of the knots. Posterior samples are obtained using a parallel tempered Metropolis-within-Gibbs Markov chain Monte Carlo algorithm. We conduct a simulation study to demonstrate that for complicated spectral densities, the B-spline prior provides more accurate Monte Carlo estimates in terms of $L_1$-error and uniform coverage probabilities than the Bernstein polynomial prior. Finally, we demonstrate the algorithm's ability to estimate a spectral density with sharp features, using real gravitational wave detector data from LIGO's sixth science run.