Correlation priors

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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
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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 & 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. 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., (2005).
[7]	M. Orispää and M. Lehtinen, Fortran linear inverse problem solver,, Inverse Problems and Imaging, 4 (2010), 482.
[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.
[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. doi: 10.1016/j.sigpro.2004.09.016.

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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. 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., (2005).
[7]	M. Orispää and M. Lehtinen, Fortran linear inverse problem solver,, Inverse Problems and Imaging, 4 (2010), 482.
[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.
[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. doi: 10.1016/j.sigpro.2004.09.016.

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