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
References:
[1]

J. M. Bardsley, Gaussian Markov random field priors for inverse problems, Inverse Problems and Imaging, 7 (2013), 397-416.  doi: 10.3934/ipi.2013.7.397.  Google Scholar

[2]

V. I. Bogachev, Gaussian Measures, American Mathematical Society, Providence RI, 1998. doi: 10.1090/surv/062.  Google Scholar

[3]

V. I. Bogachev, Measure Theory, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[4]

D. Bolin, Spatial matérn fields driven by non-gaussian noise, Scandinavian Journal of Statistics, 41 (2014), 557-579.  doi: 10.1111/sjos.12046.  Google Scholar

[5]

R. Buckdahn and E. Pardoux, Monotonicity methods for white noise driven quasi-linear SPDEs, Diffusion Processes and Related Problems in Analysis, Vol. I (Evanston, IL, 1989), Birkhäuser Boston, Boston, MA, 22 (1990), 219–233. doi: 10.1007/978-1-4684-0564-4_13.  Google Scholar

[6]

D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing – Ten Lectures on Subjective Computing, Springer, New York, 2007.  Google Scholar

[7]

D. Calvetti and E. Somersalo, A Gaussian hypermodel to recover blocky objects, Inverse Problems, 23 (2007), 733-754.  doi: 10.1088/0266-5611/23/2/016.  Google Scholar

[8]

D. Calvetti and E. Somersalo, Hypermodels in the Bayesian imaging framework, Inverse Problems, 24 (2008), 034013, 20pp. doi: 10.1088/0266-5611/24/3/034013.  Google Scholar

[9]

N. K. Chada, M. A. Iglesias, K. Roininen and A. M. Stuart, Parameterizations for Ensemble Kalman Inversion, arXiv: 1709.01781, (2016). Google Scholar

[10]

J. Charrier, Strong and weak error estimates for elliptic partial differential equations with random coefficients, SIAM Journal on Numerical Analysis, 50 (2012), 216-246.  doi: 10.1137/100800531.  Google Scholar

[11]

S. L. CotterG. O. RobertsA. M. Stuart and D. White, MCMC Methods for Functions: Modifying old algorithms to make them faster, Statistical Science, 28 (2013), 424-446.  doi: 10.1214/13-STS421.  Google Scholar

[12]

T. A. Davis, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization, ACM Trans. Math. Software, 38 (2011), Art. 8, 22 pp. doi: 10.1145/2049662.2049670.  Google Scholar

[13]

M. Dunlop, Analysis and Computation for Bayesian Inverse Problems, PhD thesis, University of Warwick, 2016. Google Scholar

[14]

M. M. Dunlop, M Girolami, A. M. Stuart and A. L. Teckentrup, How Deep Are Deep Gaussian Processes?, arXiv: 1711.11280 (2017). Google Scholar

[15]

M. Filippone and M. Girolami, Pseudo-Marginal Bayesian Inference for Gaussian Processes, IEEE Transactions Pattern Analysis and Machine Intelligence, 36 (2014), 2214-2226.   Google Scholar

[16]

L. V. Foster and T. A. Davis, Reliable calculation of numerical rank, null space bases, pseudoinverse solutions and basic solutions using SuiteSparseQR, ACM Trans. Math. Software, 40 (2013), Art. 7, 23 pp. doi: 10.1145/2513109.2513116.  Google Scholar

[17]

G.-A. FuglstadD. SimpsonF. Lindgren and H. Rue, Does non-stationary spatial data always require non-stationary random fields?, Spatial Statistics, 14 (2015), 505-531.  doi: 10.1016/j.spasta.2015.10.001.  Google Scholar

[18]

T. Helin and M. Lassas, Hierarchical models in statistical inverse problems and the Mumford-Shah functional, Inverse Problems, 27 (2011), 015008, 32pp. doi: 10.1088/0266-5611/27/1/015008.  Google Scholar

[19]

B. S. Jovanović and E. Süli, Analysis of Finite Difference Schemes, Springer, London, 2014. doi: 10.1007/978-1-4471-5460-0.  Google Scholar

[20]

J. P. Kahane, Some Random Series of Functions, Cambridge University Press, Cambridge, 1985.  Google Scholar

[21]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Springer-Verlag, New York, 2005.  Google Scholar

[22]

S. Lasanen, Non-Gaussian statistical inverse problems. Part Ⅰ: Posterior distributions, Inverse Probl. Imaging, 6 (2012), 215-266.  doi: 10.3934/ipi.2012.6.215.  Google Scholar

[23]

M. LassasE. Saksman and S. Siltanen, Discretization invariant Bayesian inversion and Besov space priors, Inverse Problems and Imaging, 3 (2009), 87-122.  doi: 10.3934/ipi.2009.3.87.  Google Scholar

[24]

M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?, Inverse Problems, 20 (2004), 1537-1563.  doi: 10.1088/0266-5611/20/5/013.  Google Scholar

[25]

F. LindgrenH. Rue and J. Lindström, An explicit link between Gaussian Markov random fields: The stochastic partial differential equation approach, Journal of the Royal Statistical Society: Series B, 73 (2011), 423-498.  doi: 10.1111/j.1467-9868.2011.00777.x.  Google Scholar

[26]

M. Markkanen, L. Roininen, J. M. J. Huttunen and S. Lasanen, Cauchy difference priors for edge-preserving Bayesian inversion with an application to x-ray tomography, arXiv: 1603.06135, (2016). Google Scholar

[27]

D. MitreaM. Mitrea and L. Yan, Boundary value problems for the Laplacian in convex and semiconvex domains, J. Funct. Anal., 258 (2010), 2507-2585.  doi: 10.1016/j.jfa.2010.01.012.  Google Scholar

[28]

J. NorbergL. RoininenJ. VierinenO. AmmD. McKay-Bukowski and M.S Lehtinen, Ionospheric tomography in Bayesian framework with Gaussian Markov random field priors, Radio Science, 50 (2015), 138-152.   Google Scholar

[29]

C. J. Paciorek, Nonstationary Gaussian Processes for Regression and Spatial Modelling, PhD thesis, Carnegie Mellon University, 2003.  Google Scholar

[30]

C. J. Paciorek and M. J. Schervish, Spatial modelling using a new class of nonstationary covariance functions, Environmetrics, 17 (2006), 483-506.  doi: 10.1002/env.785.  Google Scholar

[31]

O. PapaspiliopoulosG. O. Roberts and M. Sköld, A general framework for the parametrization of hierarchical models, Statistical Science, 22 (2007), 59-73.  doi: 10.1214/088342307000000014.  Google Scholar

[32]

L. RoininenJ. Huttunen and S. Lasanen, Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography, Inverse Problems and Imaging, 8 (2014), 561-586.  doi: 10.3934/ipi.2014.8.561.  Google Scholar

[33]

L. RoininenP. Piiroinen and M. Lehtinen, Constructing continuous stationary covariances as limits of the second-order stochastic difference equations, Inverse Problems and Imaging, 7 (2013), 611-647.  doi: 10.3934/ipi.2013.7.611.  Google Scholar

[34]

J. A. Rozanov, Markovian random fields, and stochastic partial differential equations, Mat. Sb. (N.S.), 103 (1977), 590-613.   Google Scholar

[35]

H. Rue and L. Held, Gaussian Markov Random Fields: Theory and Applications, Chapman & Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9780203492024.  Google Scholar

[36]

H. RueS. Martino and N. Chopin, Approximate Bayesian inference for latent Gaussian Models by using integrated nested Laplace approximations, Journal of the Royal Statistical Society: Series b (Statistical Methodology), 71 (2009), 319-392.  doi: 10.1111/j.1467-9868.2008.00700.x.  Google Scholar

[37]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.  Google Scholar

[38]

T. J. Sullivan, Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors, Inverse Problems and Imaging, 11 (2017), 857-874.  doi: 10.3934/ipi.2017040.  Google Scholar

[39]

J. Vanhatalo and A. Vehtari, Sparse log gaussian processes via MCMC for spatial epidemiology, JMLR Workshop and Conference Proceedings, 1 (2007), 73-89.   Google Scholar

[40]

J. Wallin and D. Bolin, Geostatistical modelling using non-Gaussian Matérn fields, Scand. J. Statist., 42 (2015), 872-890.  doi: 10.1111/sjos.12141.  Google Scholar

[41]

Y. R. YueD. SimpsonF. Lindgren and H. Rue, Bayesian adaptive smoothing splines using stochastic differential equations, Bayesian Analysis, 9 (2014), 397-423.  doi: 10.1214/13-BA866.  Google Scholar

show all references

References:
[1]

J. M. Bardsley, Gaussian Markov random field priors for inverse problems, Inverse Problems and Imaging, 7 (2013), 397-416.  doi: 10.3934/ipi.2013.7.397.  Google Scholar

[2]

V. I. Bogachev, Gaussian Measures, American Mathematical Society, Providence RI, 1998. doi: 10.1090/surv/062.  Google Scholar

[3]

V. I. Bogachev, Measure Theory, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[4]

D. Bolin, Spatial matérn fields driven by non-gaussian noise, Scandinavian Journal of Statistics, 41 (2014), 557-579.  doi: 10.1111/sjos.12046.  Google Scholar

[5]

R. Buckdahn and E. Pardoux, Monotonicity methods for white noise driven quasi-linear SPDEs, Diffusion Processes and Related Problems in Analysis, Vol. I (Evanston, IL, 1989), Birkhäuser Boston, Boston, MA, 22 (1990), 219–233. doi: 10.1007/978-1-4684-0564-4_13.  Google Scholar

[6]

D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing – Ten Lectures on Subjective Computing, Springer, New York, 2007.  Google Scholar

[7]

D. Calvetti and E. Somersalo, A Gaussian hypermodel to recover blocky objects, Inverse Problems, 23 (2007), 733-754.  doi: 10.1088/0266-5611/23/2/016.  Google Scholar

[8]

D. Calvetti and E. Somersalo, Hypermodels in the Bayesian imaging framework, Inverse Problems, 24 (2008), 034013, 20pp. doi: 10.1088/0266-5611/24/3/034013.  Google Scholar

[9]

N. K. Chada, M. A. Iglesias, K. Roininen and A. M. Stuart, Parameterizations for Ensemble Kalman Inversion, arXiv: 1709.01781, (2016). Google Scholar

[10]

J. Charrier, Strong and weak error estimates for elliptic partial differential equations with random coefficients, SIAM Journal on Numerical Analysis, 50 (2012), 216-246.  doi: 10.1137/100800531.  Google Scholar

[11]

S. L. CotterG. O. RobertsA. M. Stuart and D. White, MCMC Methods for Functions: Modifying old algorithms to make them faster, Statistical Science, 28 (2013), 424-446.  doi: 10.1214/13-STS421.  Google Scholar

[12]

T. A. Davis, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization, ACM Trans. Math. Software, 38 (2011), Art. 8, 22 pp. doi: 10.1145/2049662.2049670.  Google Scholar

[13]

M. Dunlop, Analysis and Computation for Bayesian Inverse Problems, PhD thesis, University of Warwick, 2016. Google Scholar

[14]

M. M. Dunlop, M Girolami, A. M. Stuart and A. L. Teckentrup, How Deep Are Deep Gaussian Processes?, arXiv: 1711.11280 (2017). Google Scholar

[15]

M. Filippone and M. Girolami, Pseudo-Marginal Bayesian Inference for Gaussian Processes, IEEE Transactions Pattern Analysis and Machine Intelligence, 36 (2014), 2214-2226.   Google Scholar

[16]

L. V. Foster and T. A. Davis, Reliable calculation of numerical rank, null space bases, pseudoinverse solutions and basic solutions using SuiteSparseQR, ACM Trans. Math. Software, 40 (2013), Art. 7, 23 pp. doi: 10.1145/2513109.2513116.  Google Scholar

[17]

G.-A. FuglstadD. SimpsonF. Lindgren and H. Rue, Does non-stationary spatial data always require non-stationary random fields?, Spatial Statistics, 14 (2015), 505-531.  doi: 10.1016/j.spasta.2015.10.001.  Google Scholar

[18]

T. Helin and M. Lassas, Hierarchical models in statistical inverse problems and the Mumford-Shah functional, Inverse Problems, 27 (2011), 015008, 32pp. doi: 10.1088/0266-5611/27/1/015008.  Google Scholar

[19]

B. S. Jovanović and E. Süli, Analysis of Finite Difference Schemes, Springer, London, 2014. doi: 10.1007/978-1-4471-5460-0.  Google Scholar

[20]

J. P. Kahane, Some Random Series of Functions, Cambridge University Press, Cambridge, 1985.  Google Scholar

[21]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Springer-Verlag, New York, 2005.  Google Scholar

[22]

S. Lasanen, Non-Gaussian statistical inverse problems. Part Ⅰ: Posterior distributions, Inverse Probl. Imaging, 6 (2012), 215-266.  doi: 10.3934/ipi.2012.6.215.  Google Scholar

[23]

M. LassasE. Saksman and S. Siltanen, Discretization invariant Bayesian inversion and Besov space priors, Inverse Problems and Imaging, 3 (2009), 87-122.  doi: 10.3934/ipi.2009.3.87.  Google Scholar

[24]

M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?, Inverse Problems, 20 (2004), 1537-1563.  doi: 10.1088/0266-5611/20/5/013.  Google Scholar

[25]

F. LindgrenH. Rue and J. Lindström, An explicit link between Gaussian Markov random fields: The stochastic partial differential equation approach, Journal of the Royal Statistical Society: Series B, 73 (2011), 423-498.  doi: 10.1111/j.1467-9868.2011.00777.x.  Google Scholar

[26]

M. Markkanen, L. Roininen, J. M. J. Huttunen and S. Lasanen, Cauchy difference priors for edge-preserving Bayesian inversion with an application to x-ray tomography, arXiv: 1603.06135, (2016). Google Scholar

[27]

D. MitreaM. Mitrea and L. Yan, Boundary value problems for the Laplacian in convex and semiconvex domains, J. Funct. Anal., 258 (2010), 2507-2585.  doi: 10.1016/j.jfa.2010.01.012.  Google Scholar

[28]

J. NorbergL. RoininenJ. VierinenO. AmmD. McKay-Bukowski and M.S Lehtinen, Ionospheric tomography in Bayesian framework with Gaussian Markov random field priors, Radio Science, 50 (2015), 138-152.   Google Scholar

[29]

C. J. Paciorek, Nonstationary Gaussian Processes for Regression and Spatial Modelling, PhD thesis, Carnegie Mellon University, 2003.  Google Scholar

[30]

C. J. Paciorek and M. J. Schervish, Spatial modelling using a new class of nonstationary covariance functions, Environmetrics, 17 (2006), 483-506.  doi: 10.1002/env.785.  Google Scholar

[31]

O. PapaspiliopoulosG. O. Roberts and M. Sköld, A general framework for the parametrization of hierarchical models, Statistical Science, 22 (2007), 59-73.  doi: 10.1214/088342307000000014.  Google Scholar

[32]

L. RoininenJ. Huttunen and S. Lasanen, Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography, Inverse Problems and Imaging, 8 (2014), 561-586.  doi: 10.3934/ipi.2014.8.561.  Google Scholar

[33]

L. RoininenP. Piiroinen and M. Lehtinen, Constructing continuous stationary covariances as limits of the second-order stochastic difference equations, Inverse Problems and Imaging, 7 (2013), 611-647.  doi: 10.3934/ipi.2013.7.611.  Google Scholar

[34]

J. A. Rozanov, Markovian random fields, and stochastic partial differential equations, Mat. Sb. (N.S.), 103 (1977), 590-613.   Google Scholar

[35]

H. Rue and L. Held, Gaussian Markov Random Fields: Theory and Applications, Chapman & Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9780203492024.  Google Scholar

[36]

H. RueS. Martino and N. Chopin, Approximate Bayesian inference for latent Gaussian Models by using integrated nested Laplace approximations, Journal of the Royal Statistical Society: Series b (Statistical Methodology), 71 (2009), 319-392.  doi: 10.1111/j.1467-9868.2008.00700.x.  Google Scholar

[37]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.  Google Scholar

[38]

T. J. Sullivan, Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors, Inverse Problems and Imaging, 11 (2017), 857-874.  doi: 10.3934/ipi.2017040.  Google Scholar

[39]

J. Vanhatalo and A. Vehtari, Sparse log gaussian processes via MCMC for spatial epidemiology, JMLR Workshop and Conference Proceedings, 1 (2007), 73-89.   Google Scholar

[40]

J. Wallin and D. Bolin, Geostatistical modelling using non-Gaussian Matérn fields, Scand. J. Statist., 42 (2015), 872-890.  doi: 10.1111/sjos.12141.  Google Scholar

[41]

Y. R. YueD. SimpsonF. Lindgren and H. Rue, Bayesian adaptive smoothing splines using stochastic differential equations, Bayesian Analysis, 9 (2014), 397-423.  doi: 10.1214/13-BA866.  Google Scholar

Figure 1.  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
Figure 2.  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
Figure 3.  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
Figure 4.  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
Figure 5.  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
Figure 6.  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
Figure 7.  Two-dimensional interpolation of block-shaped and Gaussian-shaped structures from noisy observations. A Matérn hyperprior is used in the analysis
Figure 8.  Two-dimensional interpolations with sparse data of block-shaped and Gaussian-shaped structures from noisy observations. A Matérn hyperprior is used in the analysis
Figure 9.  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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