Non-Gaussian statistical inverse problems. part II: Posterior convergence for approximated unknowns

Inverse Problems and Imaging

Volumn 6, Issue 2, 2012, Pages 267-287

  Lasanen, Sari ^a  

^a UNIVERSITY OF OULU   (Finland)

Author keywords

Bayesian methods; Convergence of measures; Measures on linear spaces; Non Gaussian distributions; Posterior distributions; Statistical inverse problems

Indexed keywords

EID: 84862736098     PISSN: 19308337     EISSN: 19308345     Source Type: Journal    
DOI: 10.3934/ipi.2012.6.267     Document Type: Article

References (35)

1. B. D'Ambrogi, S. Mäenpää and M. Markkanen, Discretization independent retrieval of atmospheric ozone profile, Geophysica, 35 (1999), 87-99.
2. P. Berti, L. Pratelli and P. Rigo, Almost sure weak convergence of random probability measures, Stochastics, 78 (2006), 91-97.
3. P. Billingsley, "Convergence of Probability Measures," John Wiley & Sons, New York-London Sydney, 1968.
4. V. I. Bogachev, "Gaussian Measures," American Mathematical Society, Providence, RI, 1998.
5. V. I. Bogachev, Measure Theory. Vol. I, II, Springer-Verlag, Berlin, 2007.
6. S. L. Cotter, M. Dashti, J. C. Robinson and A. M. Stuart, Bayesian inverse problems for functions and applications to fluid mechanics, Inverse Problems, 25 (2009), 5008-50051.
7. S. L. Cotter, M. Dashti and A. M. Stuart, Approximations of Bayesian inverse problems for PDEs, SIAM J. Numer. Anal., 48 (2010), 322-345.
8. I. Crimaldi and L. Pratelli, Convergence results for conditional expectations, Bernoulli, 11 (2005), 737-745.
9. I. Crimaldi and L. Pratelli, Two inequalities for conditional expectations and convergence results for filters, Statist. Probab. Lett., 74 (2005), 151-162.
10. R. M. Dudley, "Real Analysis and Probability," Cambridge University Press, Cambridge, 2002.
11. B. G. Fitzpatrick, Bayesian analysis in inverse problems, Inverse Problems, 7 (1991), 675-702.
12. V. P. Fonf, W. B. Johnson, G. Pisier and D. Preiss, Stochastic approximation properties in Banach spaces, Studia Math., 159 (2003), 103-119.
13. E. Goggin, Convergence in distribution of conditional expectations, Ann. Probab., 22 (1994), 1097-1114.
14. P. Gänssler and J. Pfanzagl, Convergence of conditional expectations, Ann. Math. Statist., 42 (1971), 315-324.
15. J. M. Hammersley, Monte Carlo methods for solving multivariable problems, Ann. New York Acad. Sci., 86 (1960), 844-874.
16. T. Helin, On infinite-dimensional hierarchical probability models in statistical inverse problems, Inverse Probl. Imaging, 3 (2009), 567-597.
17. T. Helin and M. Lassas, Hierarchical models in statistical inverse problems and the Mumford Shah functional, Inverse Problems, 27 (2011), 015008-014039.
18. W. Herer, Stochastic bases in Fréchet spaces, Demonstratio Math., 14 (1981), 719-724.
19. G. Kallianpur, "Stochastic Filtering Theory," Springer-Verlag, New York-Berlin, 1980.
20. G. Kallianpur and C. Striebel, Estimation of stochastic systems: Arbitrary system process with additive white noise observation errors, Ann. Math. Statist., 39 (1968), 785-801
21. K. Krikkeberg, Convergence of conditional expectation operators, Theory Probab. Appl., 9 (1964), 538-549.
22. J. Kuelbs, Some results of probability measures on linear topological vector spaces with an application to Strassen's log log law, J. Funct. Anal., 14 (1973), 28-43.
23. D. Landers and L. Rogge, A generalized Martingale theorem, Z. Wahrsch. Verw. Gebiete, 23 (1972), 289-292.
24. S. Lasanen, "Discretizations of Generalized Random Variables with Applications to Inverse Problems," Dissertation, Ann. Acad. Sci. Fenn. Math. Diss., No. 130, University of Oulu, 2002.
25. S. Lasanen, Non-Gaussian statistical inverse problems. Part I: Posterior distributions, Inverse Probl. and Imaging, 6 (2012), 215-266.
26. th July, 2005.
27. M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors, Inverse Probl. Imaging, 3 (2009), 87-122.
28. M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?, Inverse Problems, 20 (2004), 1537-1563.
29. H. Niederreiter, "Random Number Generation and Quasi-Monte Carlo Methods," SIAM, Philadelphia, PA, 1992.
30. B. Oeksendal, "Stochastic Differential Equations. An Introduction with Applications," Springer-Verlag, Berlin, 2003.
31. Y. Okazaki, Stochastic basis in Fréchet space, Math. Ann., 274 (1986), 379-383.
32. P. Piiroinen, "Statistical Measurements, Experiments and Applications," Dissertation, Ann. Acad. Sci. Fenn. Math. Diss., No. 143, University of Helsinki, 2005.
33. H. Sato, An ergodic measure on a locally convex topological vector space, J. Funct. Anal., 43 (1981), 149-165.
34. A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559.
35. N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan, "Probability Distributions on Banach Spaces," Reidel Publishing Co., Dordrecht, 1987.