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*correlation priors*. In contrast to some well-known smoothness priors, they have stationary covariances. The correlation priors are given in a parametric form with two parameters: correlation power and correlation length. The first parameter is connected with our prior information on the variance of the unknown. The second parameter is our prior belief on how fast the correlation of the unknown approaches zero. Roughly speaking, the correlation length is the distance beyond which two points of the unknown may be considered independent.

The prior distribution is constructed to be essentially independent of the discretization so that the a posteriori distribution will be essentially independent of the discretization grid. The covariance of a discrete correlation prior may be formed by combining the Fisher information of a discrete white noise and different-order difference priors. This is interpreted as a combination of virtual measurements of the unknown. Closed-form expressions for the continuous limits are calculated. Also, boundary correction terms for correlation priors on finite intervals are given.

A numerical example, deconvolution with a Gaussian kernel and a correlation prior, is computed.

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, numerical differentiation and deconvolution 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.

The presented construction of Whittle-Matérn correlation functions allows both isotropic or anisotropic priors with adjustable parameters in correlation length and variance. These parameters can be used, for example, to model spatially varying structural information of unknowns.

As numerical examples, we apply the developed priors to two-dimensional electrical impedance tomography problems.

*a priori*probability distributions are often given as stochastic difference equations. We derive a certain class of stochastic partial difference equations by starting from second-order stochastic partial differential equations in one and two dimensions. We discuss discretisation schemes on uniform lattices of these stationary continuous-time stochastic processes and convergence of the discrete-time processes to the continuous-time processes. A special emphasis is given to an analytical calculation of the covariance kernels of the processes. We find a representation for the covariance kernels in a simple parametric form with controllable parameters: correlation length and variance. In the discrete-time processes the discretisation step is also given as a parameter. Therefore, the discrete-time covariances can be considered as discretisation-invariant. In the two-dimensional cases we find rotation-invariant and anisotropic representations of the difference equations and the corresponding continuous-time covariance kernels.

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