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Non-Gaussian statistical inverse problems. Part I: Posterior distributions
Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns
1. | Department of Mathematical Sciences, P.O. Box 3000, 90014 University of Oulu |
References:
[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.
doi: 10.1080/17442500600745359. |
[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), 115008-1150051.
doi: 10.1088/0266-5611/25/11/115008. |
[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.
doi: 10.1137/090770734. |
[8] |
I. Crimaldi and L. Pratelli, Convergence results for conditional expectations, Bernoulli, 11 (2005), 737-745.
doi: 10.3150/bj/1126126767. |
[9] |
I. Crimaldi and L. Pratelli, Two inequalities for conditional expectations and convergence results for filters, Statist. Probab. Lett., 74 (2005), 151-162.
doi: 10.1016/j.spl.2005.04.039. |
[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.
doi: 10.4064/sm159-1-5. |
[13] |
E. Goggin, Convergence in distribution of conditional expectations, Ann. Probab., 22 (1994), 1097-1114.
doi: 10.1214/aop/1176988743. |
[14] |
P. Gänssler and J. Pfanzagl, Convergence of conditional expectations, Ann. Math. Statist., 42 (1971), 315-324.
doi: 10.1214/aoms/1177693514. |
[15] |
J. M. Hammersley, Monte Carlo methods for solving multivariable problems, Ann. New York Acad. Sci., 86 (1960), 844-874.
doi: 10.1111/j.1749-6632.1960.tb42846.x. |
[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.
doi: 10.1088/0266-5611/27/1/015008. |
[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.
doi: 10.1016/0022-1236(73)90028-1. |
[23] |
D. Landers and L. Rogge, A generalized Martingale theorem, Z. Wahrsch. Verw. Gebiete, 23 (1972), 289-292.
doi: 10.1007/BF00532514. |
[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] |
S. Lasanen and L. Roininen, Statistical inversion with Green's priors, in "Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice," Cambridge, UK, 11-15th 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.
doi: 10.1088/0266-5611/20/5/013. |
[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.
doi: 10.1007/BF01457222. |
[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.
doi: 10.1016/0022-1236(81)90026-4. |
[34] |
A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559.
doi: 10.1017/S0962492910000061. |
[35] |
N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan, "Probability Distributions on Banach Spaces," Reidel Publishing Co., Dordrecht, 1987. |
show all references
References:
[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.
doi: 10.1080/17442500600745359. |
[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), 115008-1150051.
doi: 10.1088/0266-5611/25/11/115008. |
[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.
doi: 10.1137/090770734. |
[8] |
I. Crimaldi and L. Pratelli, Convergence results for conditional expectations, Bernoulli, 11 (2005), 737-745.
doi: 10.3150/bj/1126126767. |
[9] |
I. Crimaldi and L. Pratelli, Two inequalities for conditional expectations and convergence results for filters, Statist. Probab. Lett., 74 (2005), 151-162.
doi: 10.1016/j.spl.2005.04.039. |
[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.
doi: 10.4064/sm159-1-5. |
[13] |
E. Goggin, Convergence in distribution of conditional expectations, Ann. Probab., 22 (1994), 1097-1114.
doi: 10.1214/aop/1176988743. |
[14] |
P. Gänssler and J. Pfanzagl, Convergence of conditional expectations, Ann. Math. Statist., 42 (1971), 315-324.
doi: 10.1214/aoms/1177693514. |
[15] |
J. M. Hammersley, Monte Carlo methods for solving multivariable problems, Ann. New York Acad. Sci., 86 (1960), 844-874.
doi: 10.1111/j.1749-6632.1960.tb42846.x. |
[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.
doi: 10.1088/0266-5611/27/1/015008. |
[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.
doi: 10.1016/0022-1236(73)90028-1. |
[23] |
D. Landers and L. Rogge, A generalized Martingale theorem, Z. Wahrsch. Verw. Gebiete, 23 (1972), 289-292.
doi: 10.1007/BF00532514. |
[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] |
S. Lasanen and L. Roininen, Statistical inversion with Green's priors, in "Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice," Cambridge, UK, 11-15th 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.
doi: 10.1088/0266-5611/20/5/013. |
[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.
doi: 10.1007/BF01457222. |
[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.
doi: 10.1016/0022-1236(81)90026-4. |
[34] |
A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559.
doi: 10.1017/S0962492910000061. |
[35] |
N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan, "Probability Distributions on Banach Spaces," Reidel Publishing Co., Dordrecht, 1987. |
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