Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns
Sari Lasanen
Inverse Problems & Imaging 2012, 6(2): 267-287 doi: 10.3934/ipi.2012.6.267
In practical statistical inverse problems, one often considers only finite-dimensional unknowns and investigates numerically their posterior probabilities. As many unknowns are function-valued, it is of interest to know whether the estimated probabilities converge when the finite-dimensional approximations of the unknown are refined. In this work, the generalized Bayes formula is shown to be a powerful tool in the convergence studies. With the help of the generalized Bayes formula, the question of convergence of the posterior distributions is returned to the convergence of the finite-dimensional (or any other) approximations of the unknown. The approach allows many prior distributions while the restrictions are mainly for the noise model and the direct theory. Three modes of convergence of posterior distributions are considered -- weak convergence, setwise convergence and convergence in variation. The convergence of conditional mean estimates is also studied.
keywords: Statistical inverse problems posterior distributions Bayesian methods measures on linear spaces convergence of measures non-Gaussian distributions.
Correlation priors
Lassi Roininen Markku S. Lehtinen Sari Lasanen Mikko Orispää Markku Markkanen
Inverse Problems & Imaging 2011, 5(1): 167-184 doi: 10.3934/ipi.2011.5.167
We propose a new class of Gaussian priors, 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.
keywords: a priori distributions discretization-invariance. Statistical inversion
Non-Gaussian statistical inverse problems. Part I: Posterior distributions
Sari Lasanen
Inverse Problems & Imaging 2012, 6(2): 215-266 doi: 10.3934/ipi.2012.6.215
One approach to noisy inverse problems is to use Bayesian methods. In this work, the statistical inverse problem of estimating the probability distribution of an infinite-dimensional unknown given its noisy indirect infinite-dimensional observation is studied in the Bayesian framework. The motivation for the work arises from the fact that the Bayesian computations are usually carried out in finite-dimensional cases, while the original inverse problem is often infinite-dimensional. A good understanding of an infinite-dimensional problem is, in general, helpful in finding efficient computational approaches to the problem.
    The fundamental question of well-posedness of the infinite-dimensional statistical inverse problem is considered. In particular, it is shown that the continuous dependence of the posterior probabilities on the realizations of the observation provides a certain degree of uniqueness for the posterior distribution.
    Special emphasis is on finding tools for working with non-Gaussian noise models. Especially, the applicability of the generalized Bayes formula is studied. Several examples of explicit posterior distributions are provided.
keywords: posterior distributions measures on linear spaces non-Gaussian distributions. Statistical inverse problems Bayesian methods
Hyperpriors for Matérn fields with applications in Bayesian inversion
Lassi Roininen Mark Girolami Sari Lasanen Markku Markkanen
Inverse Problems & Imaging 2019, 13(1): 1-29 doi: 10.3934/ipi.2019001

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.

keywords: Bayesian statistical estimation inverse problems Matérn fields hypermodels convergence
Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography
Lassi Roininen Janne M. J. Huttunen Sari Lasanen
Inverse Problems & Imaging 2014, 8(2): 561-586 doi: 10.3934/ipi.2014.8.561
We study flexible and proper smoothness priors for Bayesian statistical inverse problems by using Whittle-Matérn Gaussian random fields. We review earlier results on finite-difference approximations of certain Whittle-Matérn random field in $\mathbb{R}^2$. Then we derive finite-element method approximations and show that the discrete approximations can be expressed as solutions of sparse stochastic matrix equations. Such equations are known to be computationally efficient and useful in inverse problems with a large number of unknowns.
    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.
keywords: Electrical impedance tomography Bayesian statistical inversion smoothness priors.

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