Correlation priors

Previous Article
On rational approximation methods for inverse source problems
IPI Home
This Issue
Next Article
A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data

February 2011, 5(1): 167-184. doi: 10.3934/ipi.2011.5.167

Correlation priors

Lassi Roininen ^1, , Markku S. Lehtinen ^1, , Sari Lasanen ^1, , Mikko Orispää ^2, and Markku Markkanen ^3,

1.	University of Oulu, Sodankylä Geophysical Observatory, Tähteläntie 62, FI-99600 Sodankylä, Finland, Finland, Finland
2.	Sodankylä Geophysical Observatory, University of Oulu, Tähteläntie 62, FIN-99600 Sodankylä
3.	Eigenor Corporation, Lompolontie 1, FI-99600 Sodankylä, Finland

Received April 2009 Revised October 2010 Published February 2011

Abstract
Related Papers

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.

Mathematics Subject Classification: Primary: 65Q10, 60G10; Secondary: 42A3.

Citation: Lassi Roininen, Markku S. Lehtinen, Sari Lasanen, Mikko Orispää, Markku Markkanen. Correlation priors. Inverse Problems and Imaging, 2011, 5 (1) : 167-184. doi: 10.3934/ipi.2011.5.167

References:

[1]	D. Calvetti and E. Somersalo, "Introduction to Bayesian Scientific Computing - Ten Lectures on Subjective Computing," Springer, 2007.
[2]	I. I. Gikhman and A. V. Skorokhod, "The Theory of Stochastic Processes I," Springer, 1974.
[3]	I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series and Products," Academic Press, 1965.
[4]	J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer, 2005.
[5]	J. P. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo, Inverse problems with structural prior information, Inverse Problems, 15 (1999), 713-729. doi: 10.1088/0266-5611/15/3/306.
[6]	S. Lasanen and L. Roininen, "Statistical Inversion with Green's Priors," 5th Int. Conf. Inv. Prob. Eng. Proc., Cambridge, 2005.
[7]	M. Orispää and M. Lehtinen, Fortran linear inverse problem solver, Inverse Problems and Imaging, 4 (2010), 482-503.
[8]	A. Tarantola, "Inverse Problem Theory. Methods for Data Fitting and Model Parameter Estimation," Elsevier, 1987.
[9]	A. Tarantola and B. Valette, Inverse Problems = Quest for Information, Journal of Geophysics, 50 (1982), 159-170.
[10]	R. L. Wolpert and M. S. Taqqu, Fractional Ornstein-Uhlenbeck Lévy processes and the Telecom process: Upstairs and downstairs, Signal Processing, 85 (2005), 1523-1545. doi: 10.1016/j.sigpro.2004.09.016.

show all references

References:

[1]	D. Calvetti and E. Somersalo, "Introduction to Bayesian Scientific Computing - Ten Lectures on Subjective Computing," Springer, 2007.
[2]	I. I. Gikhman and A. V. Skorokhod, "The Theory of Stochastic Processes I," Springer, 1974.
[3]	I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series and Products," Academic Press, 1965.
[4]	J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer, 2005.
[5]	J. P. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo, Inverse problems with structural prior information, Inverse Problems, 15 (1999), 713-729. doi: 10.1088/0266-5611/15/3/306.
[6]	S. Lasanen and L. Roininen, "Statistical Inversion with Green's Priors," 5th Int. Conf. Inv. Prob. Eng. Proc., Cambridge, 2005.
[7]	M. Orispää and M. Lehtinen, Fortran linear inverse problem solver, Inverse Problems and Imaging, 4 (2010), 482-503.
[8]	A. Tarantola, "Inverse Problem Theory. Methods for Data Fitting and Model Parameter Estimation," Elsevier, 1987.
[9]	A. Tarantola and B. Valette, Inverse Problems = Quest for Information, Journal of Geophysics, 50 (1982), 159-170.
[10]	R. L. Wolpert and M. S. Taqqu, Fractional Ornstein-Uhlenbeck Lévy processes and the Telecom process: Upstairs and downstairs, Signal Processing, 85 (2005), 1523-1545. doi: 10.1016/j.sigpro.2004.09.016.

[1]	Zoltán Horváth, Yunfei Song, Tamás Terlaky. Steplength thresholds for invariance preserving of discretization methods of dynamical systems on a polyhedron. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2997-3013. doi: 10.3934/dcds.2015.35.2997
[2]	Matti Lassas, Eero Saksman, Samuli Siltanen. Discretization-invariant Bayesian inversion and Besov space priors. Inverse Problems and Imaging, 2009, 3 (1) : 87-122. doi: 10.3934/ipi.2009.3.87
[3]	Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems and Imaging, 2012, 6 (2) : 215-266. doi: 10.3934/ipi.2012.6.215
[4]	Lassi Roininen, Janne M. J. Huttunen, Sari Lasanen. Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography. Inverse Problems and Imaging, 2014, 8 (2) : 561-586. doi: 10.3934/ipi.2014.8.561
[5]	Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control and Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014
[6]	Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225
[7]	Le Li, Lihong Huang, Jianhong Wu. Flocking and invariance of velocity angles. Mathematical Biosciences & Engineering, 2016, 13 (2) : 369-380. doi: 10.3934/mbe.2015007
[8]	Hitoshi Ishii, Paola Loreti, Maria Elisabetta Tessitore. A PDE approach to stochastic invariance. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 651-664. doi: 10.3934/dcds.2000.6.651
[9]	Dawan Mustafa, Bernt Wennberg. Chaotic distributions for relativistic particles. Kinetic and Related Models, 2016, 9 (4) : 749-766. doi: 10.3934/krm.2016014
[10]	Axel Heim, Vladimir Sidorenko, Uli Sorger. Computation of distributions and their moments in the trellis. Advances in Mathematics of Communications, 2008, 2 (4) : 373-391. doi: 10.3934/amc.2008.2.373
[11]	Fengwei Li, Qin Yue, Fengmei Liu. The weight distributions of constacyclic codes. Advances in Mathematics of Communications, 2017, 11 (3) : 471-480. doi: 10.3934/amc.2017039
[12]	Ovidiu Carja, Victor Postolache. A Priori estimates for solutions of differential inclusions. Conference Publications, 2011, 2011 (Special) : 258-264. doi: 10.3934/proc.2011.2011.258
[13]	Jacky Cresson, Bénédicte Puig, Stefanie Sonner. Stochastic models in biology and the invariance problem. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2145-2168. doi: 10.3934/dcdsb.2016041
[14]	Adriano Da Silva, Christoph Kawan. Invariance entropy of hyperbolic control sets. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 97-136. doi: 10.3934/dcds.2016.36.97
[15]	Xing-Fu Zhong. Variational principles of invariance pressures on partitions. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 491-508. doi: 10.3934/dcds.2020019
[16]	Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3653-3666. doi: 10.3934/dcdsb.2018309
[17]	Christoph Kawan. Upper and lower estimates for invariance entropy. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 169-186. doi: 10.3934/dcds.2011.30.169
[18]	Robert Jarrow, Philip Protter, Jaime San Martin. Asset price bubbles: Invariance theorems. Frontiers of Mathematical Finance, 2022, 1 (2) : 161-188. doi: 10.3934/fmf.2021006
[19]	H.T. Banks, Jimena L. Davis. Quantifying uncertainty in the estimation of probability distributions. Mathematical Biosciences & Engineering, 2008, 5 (4) : 647-667. doi: 10.3934/mbe.2008.5.647
[20]	BronisŁaw Jakubczyk, Wojciech Kryński. Vector fields with distributions and invariants of ODEs. Journal of Geometric Mechanics, 2013, 5 (1) : 85-129. doi: 10.3934/jgm.2013.5.85

2021 Impact Factor: 1.483

Tools

Metrics

Other articles
by authors

on AIMS
Lassi Roininen Markku S. Lehtinen Sari Lasanen Mikko Orispää Markku Markkanen
on Google Scholar
Lassi Roininen Markku S. Lehtinen Sari Lasanen Mikko Orispää Markku Markkanen

[Back to Top]