  We introduce a highly efficient fully Bayesian approach for anisotropic
multidimensional smoothing. The main challenge in this context is the Markov
chain Monte Carlo update of the smoothing parameters as their full conditional
posterior comprises a pseudo-determinant that appears to be intractable at
first sight. As a consequence, most existing implementations are
computationally feasible only for the estimation of two-dimensional tensor
product smooths, which is, however, too restrictive for many applications. In
this paper, we break this barrier and derive closed-form expressions for the
log-pseudo-determinant and its first and second order partial derivatives.
These expressions are valid for arbitrary dimension and very efficient to
evaluate, which allows us to set up an efficient MCMC sampler with adaptive
Metropolis-Hastings updates for the smoothing parameters. We investigate
different priors for the smoothing parameters and discuss the efficient
derivation of lower-dimensional effects such as one-dimensional main effects
and two-dimensional interactions. We show that the suggested approach
outperforms previous suggestions in the literature in terms of accuracy,
scalability and computational cost and demonstrate its applicability by
consideration of an illustrating temperature data example from spatio-temporal
statistics.
