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Correlation priors
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 |
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.
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. |
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