Adaptive inference over Besov spaces in the white noise model using $p$-exponential priors

Abstract

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.

References in corpus (6)

• The pseudo-marginal approach for efficient Monte Carlo computations
• A General Framework for the Parametrization of Hierarchical Models
• Convergence rates of posterior distributions for noniid observations
• Reproducing kernel Hilbert spaces of Gaussian priors
• Nonparametric Bayesian inference for reversible multi-dimensional diffusions
• Laplace priors and spatial inhomogeneity in Bayesian inverse problems

Cited by in corpus (1)

• Bayesian decomposition using Besov priors