# American Institute of Mathematical Sciences

February  2019, 13(1): 1-29. doi: 10.3934/ipi.2019001

## Hyperpriors for Matérn fields with applications in Bayesian inversion

 1 LUT University, School of Engineering Science, P.O.Box 20, FI-53851 Lappeenranta, Finland 2 Department of Mathematics, Imperial College London, and Alan Turing Institute, London, United Kingdom 3 LUT University, School of Engineering Science, Finland 4 Eigenor Corporation, Lompolontie 1, FI-99600 Sodankylä, Finland

Received  December 2016 Revised  April 2017 Published  December 2018

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.

Citation: Lassi Roininen, Mark Girolami, Sari Lasanen, Markku Markkanen. Hyperpriors for Matérn fields with applications in Bayesian inversion. Inverse Problems & Imaging, 2019, 13 (1) : 1-29. doi: 10.3934/ipi.2019001
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##### References:
Examples of constructing non-stationary Matérn realisations with hypermodels. Top panel - from left to right: Realisation $\ell^N_\omega$ given Cauchy walk as $u^N_\omega$, resulting covariance matrix, and four realisations. Bottom panel: Same as above, but with a Gaussian process hyperprior
Non-stationary structures obtained by starting from a constant-parameter or inhomogeneous Matérn field realisation (upper panel), after which have been mapped to correlation length-scaling fields (middle). In the bottom panel, we have corresponding realisations with isotropic and anisotropic structures. This kind of structure can detect regions within which the behaviour of the random field is smooth, but the regions are distinct
Top panel: 81 noisy measurements and estimated $\ell^N$ ($N = 161$) with Cauchy noise (B) and Gaussian hyperprior (C). (D, G, J) are conditional mean estimates of $v^N$ ($N = 161$) with long length-scaling (D), $\ell^N$ minimising MAE (G), $\ell^N$ minimising RMSE (J). (E, H, K) and (F, I, L) are CM-estimates of $v^N$ on different meshes with Cauchy hypermodel and Gaussian hypermodels, respectively
Estimates of $\ell^N$ and $v^N$ with a Cauchy walk hypermodel $u^N$ on different lattices with $81$ measurements, with the number of unknowns varying as in figures. Bottom four subfigures (G-J) are chains and cumulative means of certain $\ell^N$ and $v^N$ elements
Estimates of $\ell^N$ and $v^N$ with a Gaussian hyperprior $u^N$ on different lattices with $81$ measurements, with the number of unknowns varying as in figures. Bottom four subfigures (G-J) are chains and cumulative means of certain $\ell^N$ and $v^N$ elements
Numerical differentiation of a noisy signal with the developed Gaussian hypermodel. We plot $v^N$ on different meshes for seeing the discretisation-invariance of the estimates
Two-dimensional interpolation of block-shaped and Gaussian-shaped structures from noisy observations. A Matérn hyperprior is used in the analysis
Two-dimensional interpolations with sparse data of block-shaped and Gaussian-shaped structures from noisy observations. A Matérn hyperprior is used in the analysis
Two-dimensional deconvolution of block-shaped and Gaussian-shaped structures from noisy observations. A Matérn hyperprior is used in the analysis
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