# American Institue of Mathematical Sciences

2017, 11(5): 857-874. doi: 10.3934/ipi.2017040

## Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors

 Free University of Berlin and Zuse Institute Berlin, Takustraße 7,14195 Berlin, Germany

Received  May 2016 Revised  November 2016 Published  July 2017

This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451–559,2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen–Loéve expansion for square-integrable random variables can be used to sample such measures on quasi-Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger metric upon perturbations of the misfit function and observed data.

Citation: T. J. Sullivan. Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors. Inverse Problems & Imaging, 2017, 11 (5) : 857-874. doi: 10.3934/ipi.2017040
##### References:
 [1] A. Achim, P. Tsakalides, A. Bezerianos, SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling, IEEE Trans. Geosci. Remote, 41 (2003), 1773-1784. doi: 10.1109/TGRS.2003.813488. [2] C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide Springer, Berlin, third edition, 2006. [3] V. I. Bogachev, Differentiable Measures and the Malliavin Calculus volume 164 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2010. [4] R. Bonic, Some properties of Hilbert scales, Proc. Amer. Math. Soc., 18 (1967), 1000-1003. doi: 10.1090/S0002-9939-1967-0230115-3. [5] J. M. Chambers, C. L. Mallows, B. W. Stuck, A method for simulating stable random variables, J. Amer. Statist. Assoc., 71 (1976), 340-344. doi: 10.1080/01621459.1976.10480344. [6] O. Christensen, D. T. Stoeva, p-frames in separable Banach spaces, Adv. Comput. Math., 18 (2003), 117-126. doi: 10.1023/A:1021364413257. [7] M. Dashti, A. M. Stuart, The Bayesian approach to inverse problems, Handbook of Uncertainty Quantification, (2016), 311-428. doi: 10.1007/978-3-319-11259-6_7-1. [8] M. Dashti, S. Harris, A. M. Stuart, Besov priors for Bayesian inverse problems, Inverse Probl. Imaging, 6 (2012), 183-200. doi: 10.3934/ipi.2012.6.183. [9] N. Hansen, F. Gemperle, A. Auger and P. Koumoutsakos, When do heavy-tail distributions help?, In T. P. Runarsson, H. -G. Beyer, E. Burke, J. J. Merelo-Guervós, L. D. Whitley, and X. Yao, editors, Parallel Problem Solving from Nature — PPSN Ⅸ: 9th International Conference, Reykjavik, Iceland, September 9–13,2006, Proceedings, Springer, Berlin, Heidelberg, (2006), 62–71. [10] B. Hosseini, N. Nigam, Well-posed Bayesian inverse problems: Priors with exponential tails, SIAM/ASA J. Uncertain. Quantif., 5 (2017), 436-465. doi: 10.1137/16M1076824. [11] J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems volume 160 of Applied Mathematical Sciences Springer-Verlag, New York, 2005. [12] C. Kraft, Some conditions for consistency and uniform consistency of statistical procedures, Univ. California Publ. Statist., 2 (1955), 125-141. [13] M. Lassas, 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. [14] M. Lassas, E. Saksman, S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors, Inverse Probl. Imaging, 3 (2009), 87-122. doi: 10.3934/ipi.2009.3.87. [15] M. Ledoux and M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes volume 23 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] Springer-Verlag, Berlin, 1991. [16] M. Markkanen, L. Roininen, J. M. J. Huttunen and S. Lasanen, Cauchy difference priors for edge-preserving Bayesian inversion with an application to X-ray tomography, 2016. arXiv: 1603. 06135v1. [17] J. P. Nolan, Stable Distributions — Models for Heavy Tailed Data, Birkhauser, Boston, 2017. In progress, Chapter 1 online at http://fs2.american.edu/jpnolan/www/stable/stable.html. [18] A. O'Hagan, Modelling with heavy tails, In Bayesian Statistics, 3 (Valencia, 1987), Oxford Sci. Publ. , Oxford Univ. Press, New York, (1988), 345–359. [19] H. Owhadi and C. Scovel, Qualitative robustness in Bayesian inference, 2016, arXiv: 1411. 3984v3. [20] M. S. Pinsker, Information and Information Stability of Random Variables and Processes Holden-Day, Inc. , San Francisco, Calif. -London-Amsterdam, 1964. [21] M. Shao, C. Nikias, Signal processing with fractional lower order moments: Stable processes and their application, Proc. IEEE, 81 (1993), 986-1010. doi: 10.1109/5.231338. [22] T. Steerneman, On the total variation and Hellinger distance between signed measures; an application to product measures, Proc. Amer. Math. Soc., 88 (1983), 684-688. doi: 10.1090/S0002-9939-1983-0702299-0. [23] A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559. doi: 10.1017/S0962492910000061. [24] A. N. Tikhonov, On the solution of incorrectly put problems and the regularisation method, In Outlines Joint Sympos. Partial Differential Equations (Novosibirsk, 1963), pages 261–265. Acad. Sci. USSR Siberian Branch, Moscow, 1963. [25] P. Tsakalides, P. Reveliotis, C. L. Nikias, Scalar quantisation of heavy-tailed signals, IEE Proc. -Vis. Image Sign., 147 (2000), 475-484. doi: 10.1049/ip-vis:20000470. [26] E. Tsionas, Monte Carlo inference in econometric models with symmetric stable distributions, J. Economet, 88 (1999), 365-401. doi: 10.1016/S0304-4076(98)00039-6.

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##### References:
 [1] A. Achim, P. Tsakalides, A. Bezerianos, SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling, IEEE Trans. Geosci. Remote, 41 (2003), 1773-1784. doi: 10.1109/TGRS.2003.813488. [2] C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide Springer, Berlin, third edition, 2006. [3] V. I. Bogachev, Differentiable Measures and the Malliavin Calculus volume 164 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2010. [4] R. Bonic, Some properties of Hilbert scales, Proc. Amer. Math. Soc., 18 (1967), 1000-1003. doi: 10.1090/S0002-9939-1967-0230115-3. [5] J. M. Chambers, C. L. Mallows, B. W. Stuck, A method for simulating stable random variables, J. Amer. Statist. Assoc., 71 (1976), 340-344. doi: 10.1080/01621459.1976.10480344. [6] O. Christensen, D. T. Stoeva, p-frames in separable Banach spaces, Adv. Comput. Math., 18 (2003), 117-126. doi: 10.1023/A:1021364413257. [7] M. Dashti, A. M. Stuart, The Bayesian approach to inverse problems, Handbook of Uncertainty Quantification, (2016), 311-428. doi: 10.1007/978-3-319-11259-6_7-1. [8] M. Dashti, S. Harris, A. M. Stuart, Besov priors for Bayesian inverse problems, Inverse Probl. Imaging, 6 (2012), 183-200. doi: 10.3934/ipi.2012.6.183. [9] N. Hansen, F. Gemperle, A. Auger and P. Koumoutsakos, When do heavy-tail distributions help?, In T. P. Runarsson, H. -G. Beyer, E. Burke, J. J. Merelo-Guervós, L. D. Whitley, and X. Yao, editors, Parallel Problem Solving from Nature — PPSN Ⅸ: 9th International Conference, Reykjavik, Iceland, September 9–13,2006, Proceedings, Springer, Berlin, Heidelberg, (2006), 62–71. [10] B. Hosseini, N. Nigam, Well-posed Bayesian inverse problems: Priors with exponential tails, SIAM/ASA J. Uncertain. Quantif., 5 (2017), 436-465. doi: 10.1137/16M1076824. [11] J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems volume 160 of Applied Mathematical Sciences Springer-Verlag, New York, 2005. [12] C. Kraft, Some conditions for consistency and uniform consistency of statistical procedures, Univ. California Publ. Statist., 2 (1955), 125-141. [13] M. Lassas, 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. [14] M. Lassas, E. Saksman, S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors, Inverse Probl. Imaging, 3 (2009), 87-122. doi: 10.3934/ipi.2009.3.87. [15] M. Ledoux and M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes volume 23 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] Springer-Verlag, Berlin, 1991. [16] M. Markkanen, L. Roininen, J. M. J. Huttunen and S. Lasanen, Cauchy difference priors for edge-preserving Bayesian inversion with an application to X-ray tomography, 2016. arXiv: 1603. 06135v1. [17] J. P. Nolan, Stable Distributions — Models for Heavy Tailed Data, Birkhauser, Boston, 2017. In progress, Chapter 1 online at http://fs2.american.edu/jpnolan/www/stable/stable.html. [18] A. O'Hagan, Modelling with heavy tails, In Bayesian Statistics, 3 (Valencia, 1987), Oxford Sci. Publ. , Oxford Univ. Press, New York, (1988), 345–359. [19] H. Owhadi and C. Scovel, Qualitative robustness in Bayesian inference, 2016, arXiv: 1411. 3984v3. [20] M. S. Pinsker, Information and Information Stability of Random Variables and Processes Holden-Day, Inc. , San Francisco, Calif. -London-Amsterdam, 1964. [21] M. Shao, C. Nikias, Signal processing with fractional lower order moments: Stable processes and their application, Proc. IEEE, 81 (1993), 986-1010. doi: 10.1109/5.231338. [22] T. Steerneman, On the total variation and Hellinger distance between signed measures; an application to product measures, Proc. Amer. Math. Soc., 88 (1983), 684-688. doi: 10.1090/S0002-9939-1983-0702299-0. [23] A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559. doi: 10.1017/S0962492910000061. [24] A. N. Tikhonov, On the solution of incorrectly put problems and the regularisation method, In Outlines Joint Sympos. Partial Differential Equations (Novosibirsk, 1963), pages 261–265. Acad. Sci. USSR Siberian Branch, Moscow, 1963. [25] P. Tsakalides, P. Reveliotis, C. L. Nikias, Scalar quantisation of heavy-tailed signals, IEE Proc. -Vis. Image Sign., 147 (2000), 475-484. doi: 10.1049/ip-vis:20000470. [26] E. Tsionas, Monte Carlo inference in econometric models with symmetric stable distributions, J. Economet, 88 (1999), 365-401. doi: 10.1016/S0304-4076(98)00039-6.
Uniform angular measure on a circle projects radially to give Cauchy measure with width parameter $\gamma$ on any line at distance $\gamma$ from the centre of the circle
Cauchy and Gaussian wavelet expansions in the linear spline orthonormal basis of $L^{2}([0, 1], \mathrm{d} x)$. Each horizontal stripe shows a random function $u(x) = \sum_{j = 0}^{J} \sum_{k = 0}^{2^{j} - 1} u_{j, k} 2^{j / 2} \psi(2^{j} x - k)$, where each $u_{j , k} = (j + 1)^{-2} 2^{-j}$ times a standard Cauchy or normal draw, and $\psi$ denotes the mother wavelet. The plots show $20$ i.i.d. samples with $J = 10$. Theorem 3.4 ensures a.s. convergence in $L^{2}([0, 1])$ as $J \to \infty$. To enable easy comparisons between plots, the ensemble has been translated and linearly scaled to take values $u(x) \in [0, 1]$, and the same random seed is used in each case
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