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

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Non-Gaussian statistical inverse problems. Part I: Posterior distributions

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

Sari Lasanen ^1,

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 and 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), 87-99.
[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 5^th International Conference on Inverse Problems in Engineering: Theory and Practice," Cambridge, UK, 11-15^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. 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 5^th International Conference on Inverse Problems in Engineering: Theory and Practice," Cambridge, UK, 11-15^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. 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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