American Institute of Mathematical Sciences

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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

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.  Google Scholar

[2]

I. I. Gikhman and A. V. Skorokhod, "The Theory of Stochastic Processes I," Springer, 1974.  Google Scholar

[3]

I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series and Products," Academic Press, 1965.  Google Scholar

[4]

J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer, 2005.  Google Scholar

[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.  Google Scholar

[6]

S. Lasanen and L. Roininen, "Statistical Inversion with Green's Priors," 5th Int. Conf. Inv. Prob. Eng. Proc., Cambridge, 2005. Google Scholar

[7]

M. Orispää and M. Lehtinen, Fortran linear inverse problem solver, Inverse Problems and Imaging, 4 (2010), 482-503. Google Scholar

[8]

A. Tarantola, "Inverse Problem Theory. Methods for Data Fitting and Model Parameter Estimation," Elsevier, 1987.  Google Scholar

[9]

A. Tarantola and B. Valette, Inverse Problems = Quest for Information, Journal of Geophysics, 50 (1982), 159-170. Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[2]

I. I. Gikhman and A. V. Skorokhod, "The Theory of Stochastic Processes I," Springer, 1974.  Google Scholar

[3]

I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series and Products," Academic Press, 1965.  Google Scholar

[4]

J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer, 2005.  Google Scholar

[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.  Google Scholar

[6]

S. Lasanen and L. Roininen, "Statistical Inversion with Green's Priors," 5th Int. Conf. Inv. Prob. Eng. Proc., Cambridge, 2005. Google Scholar

[7]

M. Orispää and M. Lehtinen, Fortran linear inverse problem solver, Inverse Problems and Imaging, 4 (2010), 482-503. Google Scholar

[8]

A. Tarantola, "Inverse Problem Theory. Methods for Data Fitting and Model Parameter Estimation," Elsevier, 1987.  Google Scholar

[9]

A. Tarantola and B. Valette, Inverse Problems = Quest for Information, Journal of Geophysics, 50 (1982), 159-170. Google Scholar

[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.  Google Scholar

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