  In many scientific applications the aim is to infer a function which is
smooth in some areas, but rough or even discontinuous in other areas of its
domain. Such spatially inhomogeneous functions can be modelled in Besov spaces
with suitable integrability parameters. In this work we study adaptive Bayesian
inference over Besov spaces, in the white noise model from the point of view of
rates of contraction, using $p$-exponential priors, which range between Laplace
and Gaussian and possess regularity and scaling hyper-parameters. To achieve
adaptation, we employ empirical and hierarchical Bayes approaches for tuning
these hyper-parameters. Our results show that, while it is known that Gaussian
priors can attain the minimax rate only in Besov spaces of spatially
homogeneous functions, Laplace priors attain the minimax or nearly the minimax
rate in both Besov spaces of spatially homogeneous functions and Besov spaces
permitting spatial inhomogeneities.
