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

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.
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 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.

[1]

Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems and 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 and 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 and 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 and 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 and 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 and 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]

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

[9]

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 and Imaging, 2020, 14 (1) : 117-132. doi: 10.3934/ipi.2019066

[10]

Mengli Hao, Ting Gao, Jinqiao Duan, Wei Xu. Non-Gaussian dynamics of a tumor growth system with immunization. Inverse Problems and 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 and 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 and 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 and 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 and 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 and Games, 2020, 7 (1) : 1-20. doi: 10.3934/jdg.2020001

[16]

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

[17]

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

[18]

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

[19]

John Maclean, Elaine T. Spiller. A surrogate-based approach to nonlinear, non-Gaussian joint state-parameter data assimilation. Foundations of Data Science, 2021, 3 (3) : 589-614. doi: 10.3934/fods.2021019

[20]

Martin Heida, Stefan Neukamm, Mario Varga. Stochastic two-scale convergence and Young measures. Networks and Heterogeneous Media, 2022, 17 (2) : 227-254. doi: 10.3934/nhm.2022004

2021 Impact Factor: 1.483

Metrics

  • PDF downloads (148)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]