  This article proposes a Bayesian approach to estimating the spectral density
of a stationary time series using a prior based on a mixture of P-spline
distributions. Our proposal is motivated by the B-spline Dirichlet process
prior of Edwards et al. (2019) in combination with Whittle's likelihood and
aims at reducing the high computational complexity of its posterior
computations. The strength of the B-spline Dirichlet process prior over the
Bernstein-Dirichlet process prior of Choudhuri et al. (2004) lies in its
ability to estimate spectral densities with sharp peaks and abrupt changes due
to the flexibility of B-splines with variable number and location of knots.
Here, we suggest to use P-splines of Eilers and Marx (1996) that combine a
B-spline basis with a discrete penalty on the basis coefficients. In addition
to equidistant knots, a novel strategy for a more expedient placement of knots
is proposed that makes use of the information provided by the periodogram about
the steepness of the spectral power distribution. We demonstrate in a
simulation study and two real case studies that this approach retains the
flexibility of the B-splines, achieves similar ability to accurately estimate
peaks due to the new data-driven knot allocation scheme but significantly
reduces the computational costs.
