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

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May 2012, 6(2): 267-287. doi: 10.3934/ipi.2012.6.267

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

Received June 2009 Revised January 2012 Published May 2012

Abstract
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In practical statistical inverse problems, one often considers only finite-dimensional unknowns and investigates numerically their posterior probabilities. As many unknowns are function-valued, it is of interest to know whether the estimated probabilities converge when the finite-dimensional approximations of the unknown are refined. In this work, the generalized Bayes formula is shown to be a powerful tool in the convergence studies. With the help of the generalized Bayes formula, the question of convergence of the posterior distributions is returned to the convergence of the finite-dimensional (or any other) approximations of the unknown. The approach allows many prior distributions while the restrictions are mainly for the noise model and the direct theory. Three modes of convergence of posterior distributions are considered -- weak convergence, setwise convergence and convergence in variation. The convergence of conditional mean estimates is also studied.

Keywords: Statistical inverse problems, posterior distributions, Bayesian methods, measures on linear spaces, convergence of measures, non-Gaussian distributions..

Mathematics Subject Classification: Primary: 60B10, 65J22; Secondary: 60B11, 62C1.

Citation: Sari Lasanen. Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems & Imaging, 2012, 6 (2) : 267-287. doi: 10.3934/ipi.2012.6.267

References:

[1]	B. D'Ambrogi, S. Mäenpää and M. Markkanen, Discretization independent retrieval of atmospheric ozone profile,, Geophysica, 35 (1999), 87. Google Scholar
[2]	P. Berti, L. Pratelli and P. Rigo, Almost sure weak convergence of random probability measures,, Stochastics, 78 (2006), 91. doi: 10.1080/17442500600745359. Google Scholar
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[7]	S. L. Cotter, M. Dashti and A. M. Stuart, Approximations of Bayesian inverse problems for PDEs,, SIAM J. Numer. Anal., 48 (2010), 322. doi: 10.1137/090770734. Google Scholar
[8]	I. Crimaldi and L. Pratelli, Convergence results for conditional expectations,, Bernoulli, 11 (2005), 737. doi: 10.3150/bj/1126126767. Google Scholar
[9]	I. Crimaldi and L. Pratelli, Two inequalities for conditional expectations and convergence results for filters,, Statist. Probab. Lett., 74 (2005), 151. doi: 10.1016/j.spl.2005.04.039. Google Scholar
[10]	R. M. Dudley, "Real Analysis and Probability,", Cambridge University Press, (2002). Google Scholar
[11]	B. G. Fitzpatrick, Bayesian analysis in inverse problems,, Inverse Problems, 7 (1991), 675. Google Scholar
[12]	V. P. Fonf, W. B. Johnson, G. Pisier and D. Preiss, Stochastic approximation properties in Banach spaces,, Studia Math., 159 (2003), 103. doi: 10.4064/sm159-1-5. Google Scholar
[13]	E. Goggin, Convergence in distribution of conditional expectations,, Ann. Probab., 22 (1994), 1097. doi: 10.1214/aop/1176988743. Google Scholar
[14]	P. Gänssler and J. Pfanzagl, Convergence of conditional expectations,, Ann. Math. Statist., 42 (1971), 315. doi: 10.1214/aoms/1177693514. Google Scholar
[15]	J. M. Hammersley, Monte Carlo methods for solving multivariable problems,, Ann. New York Acad. Sci., 86 (1960), 844. doi: 10.1111/j.1749-6632.1960.tb42846.x. Google Scholar
[16]	T. Helin, On infinite-dimensional hierarchical probability models in statistical inverse problems,, Inverse Probl. Imaging, 3 (2009), 567. Google Scholar
[17]	T. Helin and M. Lassas, Hierarchical models in statistical inverse problems and the Mumford-Shah functional,, Inverse Problems, 27 (2011), 015008. doi: 10.1088/0266-5611/27/1/015008. Google Scholar
[18]	W. Herer, Stochastic bases in Fréchet spaces,, Demonstratio Math., 14 (1981), 719. Google Scholar
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[21]	K. Krikkeberg, Convergence of conditional expectation operators,, Theory Probab. Appl., 9 (1964), 538. Google Scholar
[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. doi: 10.1016/0022-1236(73)90028-1. Google Scholar
[23]	D. Landers and L. Rogge, A generalized Martingale theorem,, Z. Wahrsch. Verw. Gebiete, 23 (1972), 289. doi: 10.1007/BF00532514. Google Scholar
[24]	S. Lasanen, "Discretizations of Generalized Random Variables with Applications to Inverse Problems," Dissertation,, Ann. Acad. Sci. Fenn. Math. Diss., (2002). Google Scholar
[25]	S. Lasanen, Non-Gaussian statistical inverse problems. Part I: Posterior distributions,, Inverse Probl. and Imaging, 6 (2012), 215. Google Scholar
[26]	S. Lasanen and L. Roininen, Statistical inversion with Green's priors,, in, (2005), 11. Google Scholar
[27]	M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors,, Inverse Probl. Imaging, 3 (2009), 87. Google Scholar
[28]	M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?,, Inverse Problems, 20 (2004), 1537. doi: 10.1088/0266-5611/20/5/013. Google Scholar
[29]	H. Niederreiter, "Random Number Generation and Quasi-Monte Carlo Methods,", SIAM, (1992). Google Scholar
[30]	B. Oeksendal, "Stochastic Differential Equations. An Introduction with Applications,", Springer-Verlag, (2003). Google Scholar
[31]	Y. Okazaki, Stochastic basis in Fréchet space,, Math. Ann., 274 (1986), 379. doi: 10.1007/BF01457222. Google Scholar
[32]	P. Piiroinen, "Statistical Measurements, Experiments and Applications," Dissertation,, Ann. Acad. Sci. Fenn. Math. Diss., (2005). Google Scholar
[33]	H. Sato, An ergodic measure on a locally convex topological vector space,, J. Funct. Anal., 43 (1981), 149. doi: 10.1016/0022-1236(81)90026-4. Google Scholar
[34]	A. M. Stuart, Inverse problems: A Bayesian perspective,, Acta Numerica, 19 (2010), 451. doi: 10.1017/S0962492910000061. Google Scholar
[35]	N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan, "Probability Distributions on Banach Spaces,", Reidel Publishing Co., (1987). Google Scholar

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References:

[1]	B. D'Ambrogi, S. Mäenpää and M. Markkanen, Discretization independent retrieval of atmospheric ozone profile,, Geophysica, 35 (1999), 87. Google Scholar
[2]	P. Berti, L. Pratelli and P. Rigo, Almost sure weak convergence of random probability measures,, Stochastics, 78 (2006), 91. doi: 10.1080/17442500600745359. Google Scholar
[3]	P. Billingsley, "Convergence of Probability Measures,", John Wiley & Sons, (1968). Google Scholar
[4]	V. I. Bogachev, "Gaussian Measures,", American Mathematical Society, (1998). Google Scholar
[5]	V. I. Bogachev, "Measure Theory. Vol. I, II,", Springer-Verlag, (2007). Google Scholar
[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. doi: 10.1088/0266-5611/25/11/115008. Google Scholar
[7]	S. L. Cotter, M. Dashti and A. M. Stuart, Approximations of Bayesian inverse problems for PDEs,, SIAM J. Numer. Anal., 48 (2010), 322. doi: 10.1137/090770734. Google Scholar
[8]	I. Crimaldi and L. Pratelli, Convergence results for conditional expectations,, Bernoulli, 11 (2005), 737. doi: 10.3150/bj/1126126767. Google Scholar
[9]	I. Crimaldi and L. Pratelli, Two inequalities for conditional expectations and convergence results for filters,, Statist. Probab. Lett., 74 (2005), 151. doi: 10.1016/j.spl.2005.04.039. Google Scholar
[10]	R. M. Dudley, "Real Analysis and Probability,", Cambridge University Press, (2002). Google Scholar
[11]	B. G. Fitzpatrick, Bayesian analysis in inverse problems,, Inverse Problems, 7 (1991), 675. Google Scholar
[12]	V. P. Fonf, W. B. Johnson, G. Pisier and D. Preiss, Stochastic approximation properties in Banach spaces,, Studia Math., 159 (2003), 103. doi: 10.4064/sm159-1-5. Google Scholar
[13]	E. Goggin, Convergence in distribution of conditional expectations,, Ann. Probab., 22 (1994), 1097. doi: 10.1214/aop/1176988743. Google Scholar
[14]	P. Gänssler and J. Pfanzagl, Convergence of conditional expectations,, Ann. Math. Statist., 42 (1971), 315. doi: 10.1214/aoms/1177693514. Google Scholar
[15]	J. M. Hammersley, Monte Carlo methods for solving multivariable problems,, Ann. New York Acad. Sci., 86 (1960), 844. doi: 10.1111/j.1749-6632.1960.tb42846.x. Google Scholar
[16]	T. Helin, On infinite-dimensional hierarchical probability models in statistical inverse problems,, Inverse Probl. Imaging, 3 (2009), 567. Google Scholar
[17]	T. Helin and M. Lassas, Hierarchical models in statistical inverse problems and the Mumford-Shah functional,, Inverse Problems, 27 (2011), 015008. doi: 10.1088/0266-5611/27/1/015008. Google Scholar
[18]	W. Herer, Stochastic bases in Fréchet spaces,, Demonstratio Math., 14 (1981), 719. Google Scholar
[19]	G. Kallianpur, "Stochastic Filtering Theory,", Springer-Verlag, (1980). Google Scholar
[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. Google Scholar
[21]	K. Krikkeberg, Convergence of conditional expectation operators,, Theory Probab. Appl., 9 (1964), 538. Google Scholar
[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. doi: 10.1016/0022-1236(73)90028-1. Google Scholar
[23]	D. Landers and L. Rogge, A generalized Martingale theorem,, Z. Wahrsch. Verw. Gebiete, 23 (1972), 289. doi: 10.1007/BF00532514. Google Scholar
[24]	S. Lasanen, "Discretizations of Generalized Random Variables with Applications to Inverse Problems," Dissertation,, Ann. Acad. Sci. Fenn. Math. Diss., (2002). Google Scholar
[25]	S. Lasanen, Non-Gaussian statistical inverse problems. Part I: Posterior distributions,, Inverse Probl. and Imaging, 6 (2012), 215. Google Scholar
[26]	S. Lasanen and L. Roininen, Statistical inversion with Green's priors,, in, (2005), 11. Google Scholar
[27]	M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors,, Inverse Probl. Imaging, 3 (2009), 87. Google Scholar
[28]	M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?,, Inverse Problems, 20 (2004), 1537. doi: 10.1088/0266-5611/20/5/013. Google Scholar
[29]	H. Niederreiter, "Random Number Generation and Quasi-Monte Carlo Methods,", SIAM, (1992). Google Scholar
[30]	B. Oeksendal, "Stochastic Differential Equations. An Introduction with Applications,", Springer-Verlag, (2003). Google Scholar
[31]	Y. Okazaki, Stochastic basis in Fréchet space,, Math. Ann., 274 (1986), 379. doi: 10.1007/BF01457222. Google Scholar
[32]	P. Piiroinen, "Statistical Measurements, Experiments and Applications," Dissertation,, Ann. Acad. Sci. Fenn. Math. Diss., (2005). Google Scholar
[33]	H. Sato, An ergodic measure on a locally convex topological vector space,, J. Funct. Anal., 43 (1981), 149. doi: 10.1016/0022-1236(81)90026-4. Google Scholar
[34]	A. M. Stuart, Inverse problems: A Bayesian perspective,, Acta Numerica, 19 (2010), 451. doi: 10.1017/S0962492910000061. Google Scholar
[35]	N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan, "Probability Distributions on Banach Spaces,", Reidel Publishing Co., (1987). Google Scholar

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