Subjective knowledge or objective belief? An oblique look to bayesian methods

Large-Scale Inverse Problems and Quantification of Uncertainty

Volumn , Issue , 2010, Pages 33-70

  Calvetti, D ^a   Somersalo, E ^a  

^a CASE WESTERN RESERVE UNIVERSITY   (United States)

Author keywords

Bayesian statistical formalism in solving inverse problems; Bayes formula and updating probabilities; Classical Tikhonov regularization scheme; Computed examples involving hypermodels; Concept related to subjective probability, being calibration allowing calibration; Meaning of probability expression of subjective belief; Subjective knowledge or objective belief oblique look of Bayesian methods; Subjective probability, and prior information use moot points; Subjectivity of probability, restricted to prior modeling and also likelihood; Updating scheme for covariance written, using Sherman Morrison Woodbury formula

Indexed keywords

EID: 84860311029     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9780470685853.ch3     Document Type: Chapter

References (34)

1. Arridge, S.R., Kaipio, J.P., Kolehmainen, V., Schweiger, M., Somersalo, E., Tarvainen, T. and Vauhkonen, M., 2006 Approximation errors and model reduction with an application in optical diffusion tomography. Inverse Problems 22, 175-195.
2. Bai, Z. and Golub, G. 1998 Some unusual eigevalue problems. In Lecture Notes In Computer Science 1573 Springer.
3. Bernardo, J.M. and Smith, A.F.M. 1994 Bayesian Theory. John Wiley& Sons, Ltd.
4. Brockwell, P.J. and Davis, R.A. 1991 Time Series: Theory and Methods. Springer Verlag.
5. Calvetti, D. and Somersalo, E. 2004 Introduction to Bayesian Scientific Computing - Ten Lectures on Subjective Computing, Springer Verlag.
6. Calvetti, D. and Somersalo, E. 2005 Statistical compensation of boundary clutter in image deblurring. Inverse Problems 21 1697-1714.
7. Calvetti, D. and Somersalo, E. 2008 Hypermodels in the Bayesian imaging framework. Inverse Problems 24 034013.
8. Calvetti, D., Kaipio, J.P. and Somersalo, E. 2006 Aristotelian prior boundary conditions. Int. J. Math. Comp. Sci. 1 63-81.
9. Caves, C.M., Fuchs, C.A. and Schack, R. 2002 Quantum probabilities as Bayesian probabilities. Phys. Rev. A 65 022305.
10. Caves, C.M., Fuchs, C.A. and Schack, R. 2007 Subjective probability and quantum certainty. Studies in History and Philosophy of modern Physics 38 255-274.
11. D’Agostini, G. 2003 Bayesian Reasoning in Data Analysis: A Critical Introduction. World Scientific.
12. D’Agostini, G. 2005 From observations to hypotheses. Probabilistic reasoning versus falsificationism and its statistical variations. arXiv:physics/0412148.
13. de Finetti, B. 1930 Funzione caratteristica di un fenomeno aleatorio. Mem. Acad. Naz. Lincei 4 86-133
14. de Finetti, B. 1974 Theory of Probability, Vol. 1, John Wiley & Sons, Ltd.
15. Diaconis, P. and Freedman, D. 1987 A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincare 23 397-423.
16. Donoho, D., Johnstone, I.M., Hoch, J.C. and Stern, A.S. 1992 Maximum entropy and the nearly black object. J. R. stat. Soc. Ser B 54 41.
17. Geman, S. and Geman, D. 1984 Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Machine Intell. 6 721-741.
18. Gilks, W.R., Richardson, S. and Spiegelhalter, D.J. 1996 Markov Chain Monte Carlo in Practice. CRC Press.
19. Golub, G. and Van Loan, C. 1996 Matrix Computations (3rd ed.), Johns Hopkins.
20. Hewitt, E. and Savage, L.J. 1955 Symmetric measures on Cartesian products. Trans. Amer. Math. Soc. 80 470-501.
21. Hyvarinen, A. and Oja, E. 2000 Independent Component Analysis: Algorithms and Applications, John Wiley & Sons, Ltd.
22. Jeffrey, R. 2004 Subjective Probability: The Real Thing. Cambridge University Press.
23. Kaipio, J.P. and Somersalo, E. 2002 Estimating anomalies from indirect observations. J. Comp. Phys. 181 398-406.
24. Kaipio, J.P. and Somersalo, E. 2004 Statistical and Computational Inverse Problems, Springer Verlag.
25. Kaipio, J.P. and Somersalo, E. 2007 Statistical inverse problems: discretization, model reduction and inverse crimes. J. Comp. Appl. Math. 198 493-504.
26. Laplace, P.S. 1774 Mémoire sur la probabilite des causes par les évenemensdn Laplace’s 1774 Memoir in Inverse Probability, Statistical Science 1 1986, pp. 359-363.
27. Lehikoinen, A., Finsterle, S., Voutilainen, A., Heikkinen, L.M., Vauhkonen, M. and Kaipio, J.P. 2007 Approximation errors and truncation of computational domains with application to geophysical tomography. Inverse Problems and Imaging 1 371-389.
28. Liu, J.S. 2004 Monte Carlo Strategies in Scientific Computing. Springer Verlag.
29. O’Hagan, A. and Forster, J. 2004 Kendall’s Advanced Theory of Statistics, Volume 2B: Bayesian Inference. Arnold
30. Press, S.J. and Tanur, J.M. 2001 The Subjectivity of Scientists and the Bayesian Approach. John Wiley & Sons, Ltd.
31. Seppanen, A., Heikkinen, L., Savolainen, T., Voutilainen, A., Somersalo, E. and Kaipio, J.P. 2007 An experimental evaluation of state estimation with fluid dynamical models in process tomography. Chem. Eng. Journal 127 23-30.
32. Starck, J.L., Pantin, E. and Murtagh, F. 2002 Deconvolution in astronomy: A review. Publ. Astron. Soc. of the Pacific 114 1051-1069.
33. Tarantola, A. 2005 Inverse Problem Theory (2nd ed.). SIAM.
34. Tarantola, A. 2006 Inverse problems, Popper and Bayes. Nature Physics 2 492-494.