No abstract

No abstract

We introduce non-stationary Matérn field priors with stochastic partial differential equations, and construct correlation length-scaling with hyperpriors. We model both the hyperprior and the Matérn prior as continuous-parameter random fields. As hypermodels, we use Cauchy and Gaussian random fields, which we map suitably to a desired correlation length-scaling range. For computations, we discretise the models with finite difference methods. We consider the convergence of the discretised prior and posterior to the discretisation limit. We apply the developed methodology to certain interpolation and numerical differentiation problems, and show numerically that we can make Bayesian inversion which promotes competing constraints of smoothness and edge-preservation. For computing the conditional mean estimator of the posterior distribution, we use a combination of Gibbs and Metropolis-within-Gibbs sampling algorithms.

We study Cauchy-distributed difference priors for edge-preserving Bayesian statistical inverse problems. On the contrary to the well-known total variation priors, one-dimensional Cauchy priors are non-Gaussian priors also in the discretization limit. Cauchy priors have independent and identically distributed increments. One-dimensional Cauchy and Gaussian random walks are special cases of Lévy α-stable random walks with α = 1 and α = 2, respectively. Both random walks can be written in closed-form, and as priors, they provide smoothing and edge-preserving properties. We briefly discuss also continuous and discrete Lévy α-stable random walks, and generalize the methodology to two-dimensional priors.We apply the developed algorithm to one-dimensional deconvolution and two-dimensional X-ray tomography problems. We compute conditional mean estimates with single-component Metropolis-Hastings and maximum a posteriori estimates with Gauss-Newton-type optimization method. We compare the proposed tomography reconstruction method to filtered back-projection estimate and conditional mean estimates with Gaussian and total variation priors.

No abstract