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

Sari Lasanen

doi:10.3934/ipi.2012.6.215

Article Contents

2012, Volume 6, Issue 2: 215-266. Doi: 10.3934/ipi.2012.6.215

This issue Previous Article Surveillance video processing using compressive sensing Next Article Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns

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

Sari Lasanen^1,

1.
Department of Mathematical Sciences, P.O. Box 3000, 90014 University of Oulu

Received: May 31, 2009

Revised: December 31, 2011

Published: April 2012

Abstract Related Papers Cited by

Abstract

One approach to noisy inverse problems is to use Bayesian methods. In this work, the statistical inverse problem of estimating the probability distribution of an infinite-dimensional unknown given its noisy indirect infinite-dimensional observation is studied in the Bayesian framework. The motivation for the work arises from the fact that the Bayesian computations are usually carried out in finite-dimensional cases, while the original inverse problem is often infinite-dimensional. A good understanding of an infinite-dimensional problem is, in general, helpful in finding efficient computational approaches to the problem.
The fundamental question of well-posedness of the infinite-dimensional statistical inverse problem is considered. In particular, it is shown that the continuous dependence of the posterior probabilities on the realizations of the observation provides a certain degree of uniqueness for the posterior distribution.
Special emphasis is on finding tools for working with non-Gaussian noise models. Especially, the applicability of the generalized Bayes formula is studied. Several examples of explicit posterior distributions are provided.

Keywords:

Mathematics Subject Classification: Primary: 60B10, 65J22; Secondary: 60B11, 62C10.

Citation:

Related Papers

Cited by

References

[1]	F. Abramovich, T. Sapatinas and B. W. Silverman, Wavelet thresholding via a Bayesian approach, J. R. Stat. Soc. Ser. B Stat. Methodol., 60 (1998), 725-749.doi: 10.1111/1467-9868.00151.
[2]	F. Abramovich and B. W. Silverman, Wavelet decomposition approaches to statistical inverse problems, Biometrika, 85 (1998), 115-129.doi: 10.1093/biomet/85.1.115.
[3]	F. Abramovich, T. Sapatinas and B. W. Silverman, Stochastic expansions in an overcomplete wavelet dictionary, Probab. Theory Related Fields, 117 (2000), 133-144.doi: 10.1007/s004400050268.
[4]	G. Backus, Isotropic probability measures in infinite-dimensional spaces, Proc. Nat. Acad. Sci. U.S.A., 84 (1987), 8755-8757.doi: 10.1073/pnas.84.24.8755.
[5]	A. Barron, M. J. Schervish and L. Wasserman, The consistency of posterior distributions in nonparametric problems, Ann. Statist., 27 (1999), 536-561.doi: 10.1214/aos/1018031206.
[6]	J.-M. Bernardo and A. F. M. Smith, "Bayesian Theory," Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Chichester, 1994.
[7]	A. B. Bhatt, G. Kallianpur and R. L. Karandikar, Robustness of the nonlinear filter, Stochastic Process. Appl., 81 (1999), 247-254.doi: 10.1016/S0304-4149(98)00106-9.
[8]	N. Bissantz and H. Holzmann, Statistical inference for inverse problems, Inverse Problems, 24 (2008), 034009, 17 pp.doi: 10.1088/0266-5611/24/3/034009.
[9]	V. I. Bogachev, "Gaussian Measures," Mathematical Surveys and Monographs, 62, American Mathematical Society, Providence, RI, 1998.
[10]	V. I. Bogachev, "Measure Theory," Vol. I, II, Springer-Verlag, Berlin, 2007.
[11]	V. I. Bogachev, A. V. Kolesnikov and K. V. Medvedev, Triangular transformations of measures, Sb. Math., 196 (2005), 309-335.doi: 10.1070/SM2005v196n03ABEH000882.
[12]	V. V. Buldygin, On invariant Bayesian estimators for generalized random variables, Theor. Probability Appl., 22 (1977), 172-175.doi: 10.1137/1122019.
[13]	J. P. Burgess and R. D. Mauldin, Conditional distributions and orthogonal measures, Ann. Probab., 9 (1981), 902-906.
[14]	V. D. Calhoun and T. Adali, Unmixing fMRI with independent component analysis, IEEE Engineering in Medicine and Biology Magazine, 25 (2006), 79-90.doi: 10.1109/MEMB.2006.1607672.
[15]	L. Cavalier, Inverse problems with non-compact operators, J. Statist. Plann. Inference, 136 (2006), 390-400.doi: 10.1016/j.jspi.2004.06.063.
[16]	L. Cavalier, Nonparametric statistical inverse problems, Inverse Problems, 24 (2008), 034004, 19 pp.
[17]	M. A. Chitre, J. R. Potter and Ong Sim-Heng, Optimal and near-optimal signal detection in snapping shrimp dominated ambient noise, IEEE J. Ocean. Eng., 31 (2006), 497-503.doi: 10.1109/JOE.2006.875272.
[18]	N. Choudhuri, S. Ghosal and A. Roy, Bayesian methods for function estimation, "Bayesian Thinking: Modeling and Computation" (eds. D. K. Dey, et al), Handbook of Statist., 25, Elsevier/North-Holland, Amsterdam, (2005), 373-414.
[19]	E. Conte and M. Longo, Characterisation of radar clutter as a spherically invariant random process, IEE Proc. Part F, 134 (1987), 191-197.
[20]	E. Conte, M. Longo and M. Lops, Modelling and simulation of non-Rayleigh radar clutter, IEE Proc. Part F, 138 (1991), 121-130.
[21]	E. Conte and A. De Maio, Mitigation techniques for non-Gaussian sea clutter, IEEE J. Ocean. Eng., 29 (2004), 284-302.doi: 10.1109/JOE.2004.826901.
[22]	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, 43 pp.doi: 10.1088/0266-5611/25/11/115008.
[23]	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.
[24]	D. D. Cox, An analysis of Bayesian inference for nonparametric regression, Ann. Statist., 21 (1993), 903-923.doi: 10.1214/aos/1176349157.
[25]	W. B. Davenport, Jr. and W. L. Root, "An Introduction to the Theory of Random Signals and Noise," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1958.
[26]	P. Diaconis and D. Freedman, On the consistency of Bayes estimates, Ann. Statist., 14 (1986), 1-67.doi: 10.1214/aos/1176349830.
[27]	P. Diaconis and D. Freedman, Consistency of Bayes estimates for nonparametric regression: Normal theory, Bernoulli, 4 (1998), 411-444.
[28]	J. Diestel and J. J. Uhl, Jr., "Vector Measures," With a foreword by B. J. Pettis, Mathematical Surveys, No. 15, American Mathematical Society, Providence, RI, 1977.
[29]	J. Dieudonné, Un exemple d'espace normal non susceptible d'une structure uniforme d'espace complet, C. R. Acad. Sci. Paris, 209 (1939), 145-147.
[30]	J. Dieudonné, Sur le théorème de Lebesgue-Nikodym. III, Ann. Univ. Grenoble. Sect. Sci. Math. Phys. (N.S.), 23 (1948), 25-53.
[31]	J. L. Doob, Stochastic processes depending on a continuous parameter, Trans. Amer. Math. Soc., 42 (1937), 107-140.doi: 10.1090/S0002-9947-1937-1501916-1.
[32]	J. L. Doob, Stochastic processes with an integral-valued parameter, Trans. Amer. Math. Soc., 44 (1938), 87-150.doi: 10.2307/1990108.
[33]	J. L. Doob, Application of the theory of martingales, in "Le Calcul des Probabilités et ses Applications," Colloques Internationaux du Centre National de la Recherche Scientifique, no. 13, Centre National de la Recherche Scientifique, Paris, (1949), 23-27.
[34]	R. M. Dudley, "Real Analysis and Probability," Revised reprint of the 1989 original, Cambridge Studies in Advanced Mathematics, 74, Cambridge University Press, Cambridge, 2002.
[35]	S. N. Evans and P. B. Stark, Inverse problems as statistics, Inverse Problems, 18 (2002), R55-R97.doi: 10.1088/0266-5611/18/4/201.
[36]	M. D. Escobar and M. West, Bayesian density estimation and inference using mixtures, J. Amer. Statist. Assoc., 90 (1995), 577-588.doi: 10.2307/2291069.
[37]	T. S. Ferguson, Prior distributions on spaces of probability measures, Ann. Statist., 2 (1974), 615-629.
[38]	T. S. Ferguson, A Bayesian analysis of some nonparametric problems, Ann. Statist., 1 (1973), 209-230.
[39]	B. G. Fitzpatrick, Bayesian analysis in inverse problems, Inverse Problems, 7 (1991), 675-702.
[40]	J.-P. Florens, M. Mouchart and J.-M. Rolin, "Elements of Bayesian Statistics," Monographs and Textbooks in Pure and AppliedMathematics, 134, Marcel Dekker, Inc., New York, 1990.
[41]	J.-P. Florens and A. Simoni, Regularizing priors for linear inverse problems, IDEI Working paper, 621 (2010).
[42]	M. Foster, An application of the Wiener-Kolmogorov smoothing theory to matrix inversion, J. Soc. Indust. Appl. Math., 9 (1961), 387-392.
[43]	J. N. Franklin, Well-posed stochastic extensons of ill-posed linear problems, J. Math. Anal. Appl., 31 (1970), 682-716.doi: 10.1016/0022-247X(70)90017-X.
[44]	M. Fréchet, On two new chapters in the theory of probability, Math. Mag., 22 (1948), 1-12.
[45]	D. Freedman, On the asymptotic behavior of Bayes' estimates in the discrete case, Ann. Math. Statist., 34 (1963), 1386-1403.
[46]	I. M. Gel'fand and N. Ya. Vilenkin, "Generalized Functions. Vol. 4: Applications of Harmonic Analysis," Academic Press, New York-London, 1964.
[47]	J. K. Ghosh and R. V. Ramamoorthi, "Bayesian Nonparametrics," Springer Series in Statistics, Springer-Verlag, New York, 2003.
[48]	Ĭ. I. Gihman and A. V. Skorohod, "The Theory of Stochastic Processes. I," Die Grundlehren der mathematischenWissenschaften, Band 210, Springer-Verlag, New York-Heidelberg, 1974.
[49]	P. Gravel, G. Beaudoin and J. A. De Guise, A method for modeling noise in medical images, IEEE Trans Med Imaging., 23 (2004), 1221-1232.doi: 10.1109/TMI.2004.832656.
[50]	U. Grenander, Stochastic processes and statistical inference, Ark. Mat., 1 (1950), 195-277.
[51]	P. R. Halmos and L. J. Savage, Application of the Radon-Nikodym theorem to the theory of sufficient statistics, Ann. Math. Statist., 20 (1949), 225-241.doi: 10.1214/aoms/1177730032.
[52]	K. Harada and H. Saigo, The space of tempered distributions as a k-space, preprint, arXiv:1009.1429.
[53]	M. Hegland, Approximate maximum a posteriori with Gaussian process priors, Constr. Approx., 26 (2007), 205-224.doi: 10.1007/s00365-006-0661-4.
[54]	T. Helin, On infinite-dimensional hierarchical probability models in statistical inverse problems, Inverse Probl. Imaging, 3 (2009), 567-597.
[55]	T. Helin and M. Lassas, Hierarchical models in statistical inverse problems and the Mumford-Shah functional, Inverse problems, 27 (2011), 015008, 32 pp.doi: 10.1088/0266-5611/27/1/015008.
[56]	J. A. Hildebrand, Anthropogenic and natural sources of ambient noise in the ocean, Mar. Ecol. Prog. Ser., 295 (2009), 5-20.
[57]	A. Hofinger and H. K. Pikkarainen, Convergence rate for the Bayesian approach to linear inverse problems, Inverse Problems, 23 (2007), 2469-2484.doi: 10.1088/0266-5611/23/6/012.
[58]	A. Hofinger and H. K. Pikkarainen, Convergence rates for linear inverse problems in the presence of an additive normal noise, Stoch. Anal. Appl., 27 (2009), 240-257.doi: 10.1080/07362990802558295.
[59]	B. Jessen, The theory of integration in a space of an infinite number of dimensions, Acta Math., 63 (1934), 249-323.doi: 10.1007/BF02547355.
[60]	M. Jiřina, On regular conditional probabilities, Czechoslovak Math. J., 9 (1959), 445-451.
[61]	M. Jiřina, Conditional probabilities on $\sigma $-algebras with countable basis, in "Select. Transl. Math. Statist. and Probability," Vol. 2, American Mathematical Society, Providence, RI, (1962), 79-86.
[62]	I. M. Johnstone and B. W. Silverman, Speed of estimation in positron emission tomography and related inverse problems, Ann. Statist., 18 (1990), 251-280.doi: 10.1214/aos/1176347500.
[63]	J.-P. Kahane, "Some Random Series of Functions," Second edition, Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge, 1985.
[64]	T. Kailath, A view of three decades of linear filtering theory, IEEE Trans. Information Theory, IT-20 (1974), 146-181.doi: 10.1109/TIT.1974.1055174.
[65]	J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Applied Mathematical Sciences, 160, Springer-Verlag, New York, 2005.
[66]	J. Kaipio and E. Somersalo, Statistical inverse problems: Discretization, model reduction and inverse crimes, J. Comput. Appl. Math., 198 (2007), 493-504.doi: 10.1016/j.cam.2005.09.027.
[67]	S. Kakutani, On equivalence of infinite product measures, Ann. of Math. (2), 49 (1948), 214-224.doi: 10.2307/1969123.
[68]	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.
[69]	K. Karhunen, Über lineare Methoden in der Wahrscheinlichkeitsrechnung, Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys., 1947 (1947), 79 pp.
[70]	E. J. Kelly, I. S. Reed and W. L. Root, The detection of radar echoes in noise. I, II, J. Soc. Indust. Appl. Math., 8 (1960), 309-341, 481-507.
[71]	G. S. Kimeldorf and G. Wahba, A correspondence between Bayesian estimation on stochastic processes and smoothing by splines, Ann. Math. Statist., 41 (1970), 495-502.doi: 10.1214/aoms/1177697089.
[72]	V. Kolehmainen, M. Lassas, K. Niinimäki and S. Siltanen, Sparsity-promoting Bayesian inversion, preprint, 2011.
[73]	A. Kolmogoroff, "Grundbegriffe der Wahrscheinlichkeitsrechnung," Springer, Berlin, 1933.
[74]	A. Kolmogorov, Stationary sequences in Hilbert's space (Russian), Bolletin Moskovskogo Gosudarstvenogo Universiteta, Matematika, 2 (1941), 40 pp.
[75]	M. Krein, On a generalization of some investigations of G. Szegö, V. Smirnoff and A. Kolmogoroff, C. R. (Doklady) Acad. Sci. URSS (N.S.), 46 (1945), 91-94.
[76]	M. Krein, On a problem of extrapolation of A. N. Kolmogoroff, C. R. (Doklady) Acad. Sci. URSS (N. S.), 46 (1945), 306-309.
[77]	P. Krug, The conditional expectation as estimator of normally distributed random variables with values in infinitely-dimensional Banach spaces, J. Multivariate Anal., 38 (1991), 1-14.doi: 10.1016/0047-259X(91)90028-Z.
[78]	H. H. Kuo, "Gaussian Measures in Banach Spaces," Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin-New York, 1975.
[79]	E. E. Kuruoglu, W. J. Fitzgerald and P. J. W. Rayner, Near optimal detection of signals in impulsive noise modeled with a symmetric $\alpha$-stable distribution, IEEE Communications Letters, 2 (1998), 282-284.
[80]	S. Lasanen, "Discretizations of Generalized Random Variables With Applications to Inverse Problems," Dissertation, Ann. Acad. Sci. Fenn. Math. Diss., No. 130, University of Oulu, Oulu, 2002.
[81]	M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors, Inverse Probl. Imaging, 3 (2009), 87-122.
[82]	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.
[83]	M. Ledoux and M. Talagrand, "Probability in Banach Spaces. Isoperimetry and Processes," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23, Springer-Verlag, Berlin, 1991.
[84]	M. Lehtinen, B. Damtie, P. Piiroinen and M. Orispää, Perfect and almost perfect pulse compression codes for range spread radar targets, Inverse probl. Imaging, 3 (2009), 465-486.
[85]	M. Lehtinen, L. Päivärinta and E. Somersalo, Linear inverse problems for generalised random variables, Inverse Problems, 5 (1989), 599-612.
[86]	M. Lewandowski, M. Ryznar, and T. Żak, Anderson inequality is strict for Gaussian and stable measures, Proc. Amer. Math. Soc., 123 (1995), 3875-3880.doi: 10.1090/S0002-9939-1995-1264821-6.
[87]	H. Luschgy, Linear estimators and Radonifying operators, Theory Probab. Appl., 40 (1995), 167-175.doi: 10.1137/1140017.
[88]	C. Macci, On the Lebesgue decomposition of the posterior distribution with respect to the prior in regular Bayesian experiments, Statist. Probab. Lett., 26 (1996), 147-152.doi: 10.1016/0167-7152(95)00004-6.
[89]	P. K. Mandal and V. Mandrekar, A Bayes formula for Gaussian noise processes and its applications, SIAM J. Control Optim., 39 (2000), 852-871.doi: 10.1137/S0363012998343380.
[90]	A. Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space, Z. Wahrsch. Verw. Gebiete, 65 (1984), 385-397.doi: 10.1007/BF00533743.
[91]	P. Müller and F. A. Quintana, Nonparametric Bayesian data analysis, Statist. Sci., 19 (2004), 95-110.doi: 10.1214/088342304000000017.
[92]	A. Neubauer and H. K. Pikkarainen, Convergence results for the Bayesian inversion theory, J. Inverse Ill-Posed Probl., 16 (2008), 601-613.doi: 10.1515/JIIP.2008.032.
[93]	J. Neveu, "Discrete-Parameter Martingales," Revised edition, North Holland Mathematical Library, Vol. 10, North-Holland Publishing Co., Amsterdam-Oxford, American Elsevier Publishing Co., Inc., New York, 1975.
[94]	B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications," Sixth edition, Universitext, Springer-Verlag, Berlin, 2003.
[95]	F. O'Sullivan, A statistical perspective on ill-posed inverse problems, Statist. Sci., 1 (1986), 502-527.
[96]	K. R. Parthasarathy, "Probability Measures on Metric Spaces," Reprint of the 1967 original, AMS Chelsea Publishing, Providence, RI, 2005.
[97]	B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc., 44 (1938), 277-304.doi: 10.1090/S0002-9947-1938-1501970-8.
[98]	D. L. Philips, A technique for the numerical solution of certain integral equations of the first kind, Journal of the ACM, 9 (1962), 84-97.
[99]	P. Piiroinen, "Statistical Measurements, Experiments and Applications," Dissertation, Ann. Acad. Sci. Fenn. Math. Diss., No. 143, University of Helsinki, Helsinki, 2005.
[100]	H. Poincaré, "Science and Hypothesis," Walter Scott Publishing, London, 1905.
[101]	H. Poincaré, "Calcul des Probabilités," Reprint of the second (1912) edition, Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1987.
[102]	P. M. Prenter and C. R. Vogel, Stochastic inversion of linear first kind integral equations. I. Continuous theory and the stochastic generalized inverse, J. Math. Anal. Appl., 106 (1985), 202-218.doi: 10.1016/0022-247X(85)90144-1.
[103]	D. Ramachandran, A note on regular conditional probabilities in Doob's sense, Ann. Probab., 9 (1981), 907-908.
[104]	D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion," Third edition, Grudlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293, Springer-Verlag, Berlin, 1999.
[105]	C. P. Robert, "The Bayesian Choice. From Decision-Theoretic Foundations to Computational Implementation," Second edition, Springer Texts in Statistics, Springer-Verlag, New York, 2001.
[106]	V. A. Rohlin, On the fundamental ideas of measure theory (Russian), Mat. Sbornik N.S., 25(67) (1949), 107-150. Translated in Amer. Math. Soc. Translation, 71 (1952), 55 pp.
[107]	W. Rudin, Lebesgue's first theorem, in "Mathematical Analysis and Applications" (ed. L. Nachbin), Part B, Adv. in Math. Suppl. Stud., 7b, Academic Press, New York-London, (1981), 741-747.
[108]	G. Samorodnitsky and M. S. Taqqu, "Stable Non-Gaussian Random Processes," Stochastic Models with Infinite Variance, Stochastic Modeling, Chapman & Hall, New York, 1994.
[109]	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.
[110]	V. V. Sazonov, On perfect measures, Izv. Akad. Nauk SSSR Ser. Mat., 26 (1962), 391-414; Translated in American Mathematical Society Translations, Series 2, Vol. 48, Fourteen papers on logic, algebra, complex variables and topology, American Mathematical Society, Providence, RI, 1965.
[111]	M. J. Schervish, "Theory of Statistics," Springer Series in Statistics, Springer-Verlag, New York, 1995.
[112]	L. Schwartz, "Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures," Tata Institute of Fundamental Research Studies in Mathematics, No. 6, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1973.
[113]	L. Schwartz, On Bayes procedures, Z. Wahrsch. Verw. Gebiete, 4 (1965), 10-26.
[114]	H. Shimomura, Some new examples of quasi-invariant measures on a Hilbert space, Publ. Res. Inst. Math. Sci., 11 (1975/76), 635-649.
[115]	A. N. Shiryaev, "Probability," Translated from the first (1980) Russian edition by R. P. Boas, Second edition, Graduate Texts in Mathematics, 95, Springer-Verlag, New York, 1996.
[116]	A. Simoni, "Bayesian Analysis of Linear Inverse Problems with Applications in Economics and Finance," Dissertation, Univ. of Bologna, 2009.
[117]	E. Slutsky, Quelques propositions sur la théorie des fonctions aléatoires (Russian. French summary), Acta [Trudy] Univ. Asiae Mediae. Ser. V-a., 1939 (1939), 15 pp.
[118]	D. M. Steinberg, A Bayesian approach to flexible modeling of multivariable response functions, J. Multivariate Anal., 34 (1990), 157-172.doi: 10.1016/0047-259X(90)90033-E.
[119]	O. N. Strand and E. R. Westwater, Statistical estimation of the numerical solution of a Fredholm integral equation of the first kind, J. Assoc. Comput. Mach., 15 (1968), 100-114.doi: 10.1145/321439.321445.
[120]	A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559.doi: 10.1017/S0962492910000061.
[121]	A. Tarantola, "Inverse Problem Theory. Methods for Data Fitting and Model Parameter Estimation," Elsevier Science Publishers, B.V., Amsterdam, 1987.
[122]	A. Tarantola and B. Valette, Inverse Problems = Quest for Information, J. Geophys., 50 (1982), 159-170.
[123]	T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge and J. P. Kaipio, An approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography, Inverse Problems, 26 (2010), 015005, 18 pp.doi: 10.1088/0266-5611/26/1/015005.
[124]	L. Tenorio, Statistical regularization of inverse problems, SIAM Rev., 43 (2001), 347-366.doi: 10.1137/S0036144500358232.
[125]	G. E. F. Thomas, Integration of functions with values in locally convex Suslin spaces, Trans. Amer. Math. Soc., 212 (1975), 61-81.doi: 10.1090/S0002-9947-1975-0385067-1.
[126]	V. F. Turchin, Statistical regularization, in "Advanced Methods in the Evaluation of Nuclear Scattering Data" (eds. H. J. Krappe, et al) (Berlin, 1985), Lecture Notes in Phys., 236, Springer, Berlin, (1985), 33-49.
[127]	S. Twomey, On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature, J. Assoc. Comput. Mach., 10 (1963), 97-101.doi: 10.1145/321150.321157.
[128]	Y. Umemura, Measures on infinite dimensional vector spaces, Publ. Res. Inst. Math. Sci. Ser. A, 1 (1965), 1-47.
[129]	R. J. Urick, "Ambient Noise in the Sea," Undersea Warfare Technology Office, Naval Sea Systems Command, Dept. of the Navy, Washington, D.C., 1984.
[130]	N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan, "Probability Distributions on Banach Spaces," Reidel Publishing Co., Dordrecht, 1987.
[131]	A. W. van der Vaart and J. H. van Zanten, Rates of contraction of posterior distributions based on Gaussian process priors, Ann. Statist., 36 (2008), 1435-1463.doi: 10.1214/009053607000000613.
[132]	V. S. Varadarajan, "Measures on Topological Spaces," Amer. Math. Soc. Transl., 2 (1965), 161-220.
[133]	G. Wahba, Improper priors, spline smoothing and the problem of guarding against model errors in regression, J. Roy. Statist. Soc. Ser. B, 40 (1978), 364-372.
[134]	S. G. Walker, P. Damien, P. W. Laud and A. F. M. Smith, Bayesian nonparametric inference for random distributions and related functions, With discussion and a reply by the authors, J. R. Stat. Soc. Ser. B Stat. Methodol., 61 (1999), 485-527.doi: 10.1111/1467-9868.00190.
[135]	R. J. Webster, Ambient noise statistics, IEEE Trans. Signal Proces., 41 (1993), 2249-2253.doi: 10.1109/78.218152.
[136]	P. Whittle, Curve and periodogram smoothing, J. Roy. Statist. Soc. Ser. B, 19 (1957), 38-47.
[137]	P. Whittle, On the smoothing of probability density functions, J. Roy. Statist. Soc. Ser. B, 20 (1958), 334-343.
[138]	N. Wiener, "Extrapolation, Interpolation, and Smoothing of Stationary Time Series. With Engineering Applications," Chapman & Hall, Ltd., London, 1949.
[139]	N. Wiener, "Collected Works. Vol. I" (ed. P. Masani), MIT Press, Cambridge, Mass.-London, 1976.
[140]	S. Willard, "General Topology," Dover Publications Inc., Mineola, NY, 2004.
[141]	G. Wise and N. Gallagher, On spherically invariant random processes, IEEE Trans. Information theory, 24 (1978), 118-120.doi: 10.1109/TIT.1978.1055841.
[142]	R. L. Wolpert and K. Ickstadt, Reflecting uncertainty in inverse problems: A Bayesian solution using Lévy processes, Inverse Problems, 20 (2004), 1759-1771.doi: 10.1088/0266-5611/20/6/004.
[143]	R. L. Wolpert. K. Ickstadt and M. B. Hansen, A nonparametric Bayesian approach to inverse problems, in "Bayesian Statistics," 7, Oxford Univ. Press, New York, (2003), 403-417.
[144]	D. X. Xia, "Measure and Integration Theory on Infinite-Dimensional Spaces," Academic Press, New York-London, 1972.
[145]	Y. Xing and B. Ranneby, Sufficient conditions for Bayesian consistency, J. Statist. Plann. Inference, 139 (2009), 2479-2489.doi: 10.1016/j.jspi.2008.11.008.
[146]	Y. Yamasaki, "Measures on Infinite-Dimensional Spaces," World Scientific Publishing Co., Singapore, 1985.
[147]	L. H. Zhao, Bayesian aspects of some nonparametric problems, Ann. Statist., 28 (2000), 532-552.doi: 10.1214/aos/1016218229.
[148]	W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation," Springer-Verlag, New York, 1989.