American Institute of Mathematical Sciences

Advanced
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

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), 87-99. Google Scholar

[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.  Google Scholar

[3]

P. Billingsley, "Convergence of Probability Measures," John Wiley & Sons, New York-London-Sydney, 1968.  Google Scholar

[4]

V. I. Bogachev, "Gaussian Measures," American Mathematical Society, Providence, RI, 1998.  Google Scholar

[5]

V. I. Bogachev, "Measure Theory. Vol. I, II," Springer-Verlag, Berlin, 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), 115008-1150051. 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), 322-345. doi: 10.1137/090770734.  Google Scholar

[8]

I. Crimaldi and L. Pratelli, Convergence results for conditional expectations, Bernoulli, 11 (2005), 737-745. 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), 151-162. doi: 10.1016/j.spl.2005.04.039.  Google Scholar

[10]

R. M. Dudley, "Real Analysis and Probability," Cambridge University Press, Cambridge, 2002.  Google Scholar

[11]

B. G. Fitzpatrick, Bayesian analysis in inverse problems, Inverse Problems, 7 (1991), 675-702.  Google Scholar

[12]

V. P. Fonf, W. B. Johnson, G. Pisier and D. Preiss, Stochastic approximation properties in Banach spaces, Studia Math., 159 (2003), 103-119. doi: 10.4064/sm159-1-5.  Google Scholar

[13]

E. Goggin, Convergence in distribution of conditional expectations, Ann. Probab., 22 (1994), 1097-1114. doi: 10.1214/aop/1176988743.  Google Scholar

[14]

P. Gänssler and J. Pfanzagl, Convergence of conditional expectations, Ann. Math. Statist., 42 (1971), 315-324. 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), 844-874. 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), 567-597.  Google Scholar

[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.  Google Scholar

[18]

W. Herer, Stochastic bases in Fréchet spaces, Demonstratio Math., 14 (1981), 719-724.  Google Scholar

[19]

G. Kallianpur, "Stochastic Filtering Theory," Springer-Verlag, New York-Berlin, 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), 785-801  Google Scholar

[21]

K. Krikkeberg, Convergence of conditional expectation operators, Theory Probab. Appl., 9 (1964), 538-549.  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), 28-43. 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), 289-292. 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., No. 130, University of Oulu, 2002.  Google Scholar

[25]

S. Lasanen, Non-Gaussian statistical inverse problems. Part I: Posterior distributions, Inverse Probl. and Imaging, 6 (2012), 215-266. Google Scholar

[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. Google Scholar

[27]

M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors, Inverse Probl. Imaging, 3 (2009), 87-122.  Google Scholar

[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.  Google Scholar

[29]

H. Niederreiter, "Random Number Generation and Quasi-Monte Carlo Methods," SIAM, Philadelphia, PA, 1992.  Google Scholar

[30]

B. Oeksendal, "Stochastic Differential Equations. An Introduction with Applications," Springer-Verlag, Berlin, 2003.  Google Scholar

[31]

Y. Okazaki, Stochastic basis in Fréchet space, Math. Ann., 274 (1986), 379-383. doi: 10.1007/BF01457222.  Google Scholar

[32]

P. Piiroinen, "Statistical Measurements, Experiments and Applications," Dissertation, Ann. Acad. Sci. Fenn. Math. Diss., No. 143, University of Helsinki, 2005.  Google Scholar

[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.  Google Scholar

[34]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559. 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., Dordrecht, 1987.  Google Scholar

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. Google Scholar

[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.  Google Scholar

[3]

P. Billingsley, "Convergence of Probability Measures," John Wiley & Sons, New York-London-Sydney, 1968.  Google Scholar

[4]

V. I. Bogachev, "Gaussian Measures," American Mathematical Society, Providence, RI, 1998.  Google Scholar

[5]

V. I. Bogachev, "Measure Theory. Vol. I, II," Springer-Verlag, Berlin, 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), 115008-1150051. 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), 322-345. doi: 10.1137/090770734.  Google Scholar

[8]

I. Crimaldi and L. Pratelli, Convergence results for conditional expectations, Bernoulli, 11 (2005), 737-745. 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), 151-162. doi: 10.1016/j.spl.2005.04.039.  Google Scholar

[10]

R. M. Dudley, "Real Analysis and Probability," Cambridge University Press, Cambridge, 2002.  Google Scholar

[11]

B. G. Fitzpatrick, Bayesian analysis in inverse problems, Inverse Problems, 7 (1991), 675-702.  Google Scholar

[12]

V. P. Fonf, W. B. Johnson, G. Pisier and D. Preiss, Stochastic approximation properties in Banach spaces, Studia Math., 159 (2003), 103-119. doi: 10.4064/sm159-1-5.  Google Scholar

[13]

E. Goggin, Convergence in distribution of conditional expectations, Ann. Probab., 22 (1994), 1097-1114. doi: 10.1214/aop/1176988743.  Google Scholar

[14]

P. Gänssler and J. Pfanzagl, Convergence of conditional expectations, Ann. Math. Statist., 42 (1971), 315-324. 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), 844-874. 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), 567-597.  Google Scholar

[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.  Google Scholar

[18]

W. Herer, Stochastic bases in Fréchet spaces, Demonstratio Math., 14 (1981), 719-724.  Google Scholar

[19]

G. Kallianpur, "Stochastic Filtering Theory," Springer-Verlag, New York-Berlin, 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), 785-801  Google Scholar

[21]

K. Krikkeberg, Convergence of conditional expectation operators, Theory Probab. Appl., 9 (1964), 538-549.  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), 28-43. 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), 289-292. 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., No. 130, University of Oulu, 2002.  Google Scholar

[25]

S. Lasanen, Non-Gaussian statistical inverse problems. Part I: Posterior distributions, Inverse Probl. and Imaging, 6 (2012), 215-266. Google Scholar

[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. Google Scholar

[27]

M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors, Inverse Probl. Imaging, 3 (2009), 87-122.  Google Scholar

[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.  Google Scholar

[29]

H. Niederreiter, "Random Number Generation and Quasi-Monte Carlo Methods," SIAM, Philadelphia, PA, 1992.  Google Scholar

[30]

B. Oeksendal, "Stochastic Differential Equations. An Introduction with Applications," Springer-Verlag, Berlin, 2003.  Google Scholar

[31]

Y. Okazaki, Stochastic basis in Fréchet space, Math. Ann., 274 (1986), 379-383. doi: 10.1007/BF01457222.  Google Scholar

[32]

P. Piiroinen, "Statistical Measurements, Experiments and Applications," Dissertation, Ann. Acad. Sci. Fenn. Math. Diss., No. 143, University of Helsinki, 2005.  Google Scholar

[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.  Google Scholar

[34]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559. 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., Dordrecht, 1987.  Google Scholar

[1]

Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems & Imaging, 2012, 6 (2) : 215-266. doi: 10.3934/ipi.2012.6.215
[2]

Azmy S. Ackleh, Ben G. Fitzpatrick, Horst R. Thieme. Rate distributions and survival of the fittest: a formulation on the space of measures. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 917-928. doi: 10.3934/dcdsb.2005.5.917
[3]

Eugen Mihailescu. Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2485-2502. doi: 10.3934/dcds.2012.32.2485
[4]

Junxiong Jia, Jigen Peng, Jinghuai Gao. Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption. Inverse Problems & Imaging, 2021, 15 (2) : 201-228. doi: 10.3934/ipi.2020061
[5]

Todd Young. Asymptotic measures and distributions of Birkhoff averages with respect to Lebesgue measure. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 359-378. doi: 10.3934/dcds.2003.9.359
[6]

Kui Lin, Shuai Lu, Peter Mathé. Oracle-type posterior contraction rates in Bayesian inverse problems. Inverse Problems & Imaging, 2015, 9 (3) : 895-915. doi: 10.3934/ipi.2015.9.895
[7]

Fengwei Li, Qin Yue, Xiaoming Sun. The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes. Advances in Mathematics of Communications, 2021, 15 (1) : 131-153. doi: 10.3934/amc.2020049
[8]

Didi Lv, Qingping Zhou, Jae Kyu Choi, Jinglai Li, Xiaoqun Zhang. Nonlocal TV-Gaussian prior for Bayesian inverse problems with applications to limited CT reconstruction. Inverse Problems & Imaging, 2020, 14 (1) : 117-132. doi: 10.3934/ipi.2019066
[9]

Miguel-C. Muñoz-Lecanda. On some aspects of the geometry of non integrable distributions and applications. Journal of Geometric Mechanics, 2018, 10 (4) : 445-465. doi: 10.3934/jgm.2018017
[10]

Mengli Hao, Ting Gao, Jinqiao Duan, Wei Xu. Non-Gaussian dynamics of a tumor growth system with immunization. Inverse Problems & Imaging, 2013, 7 (3) : 697-716. doi: 10.3934/ipi.2013.7.697
[11]

Hongjun Gao, Fei Liang. On the stochastic beam equation driven by a Non-Gaussian Lévy process. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1027-1045. doi: 10.3934/dcdsb.2014.19.1027
[12]

Daijun Jiang, Hui Feng, Jun Zou. Overlapping domain decomposition methods for linear inverse problems. Inverse Problems & Imaging, 2015, 9 (1) : 163-188. doi: 10.3934/ipi.2015.9.163
[13]

Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems & Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183
[14]

Tan Bui-Thanh, Quoc P. Nguyen. FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems. Inverse Problems & Imaging, 2016, 10 (4) : 943-975. doi: 10.3934/ipi.2016028
[15]

Max-Olivier Hongler. Mean-field games and swarms dynamics in Gaussian and non-Gaussian environments. Journal of Dynamics & Games, 2020, 7 (1) : 1-20. doi: 10.3934/jdg.2020001
[16]

Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems & Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81
[17]

Dawan Mustafa, Bernt Wennberg. Chaotic distributions for relativistic particles. Kinetic & Related Models, 2016, 9 (4) : 749-766. doi: 10.3934/krm.2016014
[18]

Axel Heim, Vladimir Sidorenko, Uli Sorger. Computation of distributions and their moments in the trellis. Advances in Mathematics of Communications, 2008, 2 (4) : 373-391. doi: 10.3934/amc.2008.2.373
[19]

Fengwei Li, Qin Yue, Fengmei Liu. The weight distributions of constacyclic codes. Advances in Mathematics of Communications, 2017, 11 (3) : 471-480. doi: 10.3934/amc.2017039
[20]

Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (87)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences