{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:24:24Z","timestamp":1787340264984,"version":"3.56.0"},"reference-count":58,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM\/ASA J. Uncertainty Quantification"],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>We present a new class of prior measures based on the generalized Gamma distribution that are closely related to $\\ell_p$-regularization techniques when $p \\in(0,1)$. Furthermore, we use the laws of pure jump L\u00e9vy processes in order to define new classes of prior measures that are concentrated on the space of functions with bounded variation. These priors serve as an alternative to the classic total variation prior and result in well-defined inverse problems. Some of these prior measures are heavy-tailed, nonconvex, and infinitely divisible. Motivated by this observation we study the class of infinitely divisible prior measures and draw a connection between their tail behavior and the tail behavior of their L\u00e9vy measures. We then study the well-posedness of Bayesian inverse problems in a general enough setting that encompasses the above-mentioned classes of prior measures. We establish that well-posedness relies on a balance between the growth of the log-likelihood function and the tail behavior of the prior and apply our results to special cases such as additive noise models and linear problems. Finally, we discuss some of the practical aspects of Bayesian inverse problems such as their consistent approximation and present three concrete examples of well-posed Bayesian inverse problems with heavy-tailed or stochastic process prior measures.<\/jats:p>","DOI":"10.1137\/16m1096372","type":"journal-article","created":{"date-parts":[[2017,10,10]],"date-time":"2017-10-10T16:11:46Z","timestamp":1507651906000},"page":"1024-1060","source":"Crossref","is-referenced-by-count":23,"title":["Well-Posed Bayesian Inverse Problems with Infinitely Divisible and Heavy-Tailed Prior Measures"],"prefix":"10.1137","volume":"5","author":[{"given":"Bamdad","family":"Hosseini","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,10,10]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"R. J. Adler,\n                      The Geometry of Random Fields\n                      , Classics Appl. Math. 62, SIAM, Philadelphia, 2010,https:\/\/doi.org\/10.1137\/1.9780898718980.","DOI":"10.1137\/1.9780898718980"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"C. D. Aliprantis and K. Border,\n                      Infinite Dimensional Analysis: A Hitchhiker's Guide\n                      , 3rd ed., Springer Science & Business Media, New York, 2006,https:\/\/doi.org\/10.1007\/3-540-29587-9.","DOI":"10.1007\/3-540-29587-9"},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"D. Applebaum,\n                      Le\u0301vy Processes and Stochastic Calculus\n                      , Cambridge Stud. Adv. Math. 93, Cambridge University Press, Cambridge, UK, 2009,https:\/\/doi.org\/10.1017\/CBO9780511809781.","DOI":"10.1017\/CBO9780511809781"},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"J.M. Azai\u0308s and M. Wschebor,\n                      Level Sets and Extrema of Random Processes and Fields\n                      , John Wiley & Sons, Hoboken, NJ, 2009,https:\/\/doi.org\/10.1002\/9780470434642.","DOI":"10.1002\/9780470434642"},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"J. M. Bernardo and A. F. M. Smith,\n                      Bayesian Theory\n                      , Wiley Ser. Probab. Statist., John Wiley & Sons, Chichester, UK, 2000,https:\/\/doi.org\/10.1002\/9780470316870.","DOI":"10.1002\/9780470316870"},{"key":"atypb6","unstructured":"P. Billingsley,\n                      Probability and Measure\n                      , Wiley Ser. Probab. Math. Statist., 3rd ed., John Wiley & Sons, New York, 1995."},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"V. I. Bogachev,\n                      Gaussian Measures\n                      , Math. Surveys Monogr. 62, American Mathematical Society, Providence, RI, 1998,https:\/\/doi.org\/10.1090\/surv\/062.","DOI":"10.1090\/surv\/062\/03"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"V. I. Bogachev,\n                      Measure Theory\n                      , Vol. 1, Springer-Verlag, Berlin, 2007,https:\/\/doi.org\/10.1007\/978-3-540-34514-5.","DOI":"10.1007\/978-3-540-34514-5"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"V. I. Bogachev,\n                      Measure Theory\n                      , Vol. 2, Springer-Verlag, Berlin, 2007.","DOI":"10.1007\/978-3-540-34514-5"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176994890"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/BF02384761"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"G. Buttazzo, M. Giaquinta, and S. Hildebrandt,\n                      One-Dimensional Variational Problems: An Introduction\n                      , Oxford Lecture Ser. Math. Appl. 15, Oxford University Press, Oxford, UK, 1998.","DOI":"10.1093\/oso\/9780198504658.001.0001"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/30\/11\/110301"},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"D. Calvetti and E. Somersalo,\n                      An Introduction to Bayesian Scientific Computing: Ten Lectures on Subjective Computing\n                      , Surv. Tutor. Appl. Math. Sci. 2, Springer Science$+$Business Media, New York, 2007,https:\/\/doi.org\/10.1007\/978-0-387-73394-4.","DOI":"10.1007\/978-0-387-73394-4"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.21432"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/asq017"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1214\/15-AOS1334"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1214\/12-AOS1029"},{"key":"atypb19","doi-asserted-by":"crossref","unstructured":"R. Cont and P. Tankov,\n                      Financial Modelling with Jump Processes\n                      , Chapman & Hall\/CRC Financ. Math. Ser., CRC Press, Boca Raton, FL, 2004,https:\/\/doi.org\/10.1201\/9780203485217.","DOI":"10.1201\/9780203485217"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/090770734"},{"key":"atypb21","first-page":"231","author":"Dainty J. C.","year":"1987","journal-title":"Orlando, FL"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.3934\/ipi.2012.6.183"},{"key":"atypb23","first-page":"1","author":"Dashti M.","year":"2016","journal-title":"Switzerland"},{"key":"atypb24","doi-asserted-by":"crossref","unstructured":"I. Daubechies,\n                      Ten Lectures on Wavelets\n                      , CBMS-NSF Regional Conf. Ser. in Appl. Math. 61, SIAM, Philadelphia, 1992,https:\/\/doi.org\/10.1137\/1.9781611970104.","DOI":"10.1137\/1.9781611970104"},{"key":"atypb25","doi-asserted-by":"crossref","unstructured":"L. C. Evans and R. F. Gariepy,\n                      Measure Theory and Fine Properties of Functions\n                      , Textb. Math., revised ed., CRC Press, Boca Raton, FL, 2015.","DOI":"10.1201\/b18333"},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"S. Foucart and H. Rauhut,\n                      A Mathematical Introduction to Compressive Sensing\n                      , Appl. Numer. Harmon. Anal., Birkha\u0308user\/Springer, New York, 2013,https:\/\/doi.org\/10.1007\/978-0-8176-4948-7.","DOI":"10.1007\/978-0-8176-4948-7"},{"key":"atypb27","doi-asserted-by":"crossref","unstructured":"R. G. Ghanem and P. D. Spanos,\n                      Stochastic Finite Elements: A Spectral Approach\n                      , Dover, New York, 2003,https:\/\/doi.org\/10.1007\/978-1-4612-3094-6.","DOI":"10.1007\/978-1-4612-3094-6"},{"key":"atypb28","unstructured":"P. Ghosh and A. Chakrabarti,\n                      Posterior Concentration Properties of a General Class of Shrinkage Priors around Nearly Black Vectors\n                      , preprint, 2014,https:\/\/arxiv.org\/abs\/1412.8161."},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1016\/j.exmath.2010.03.001"},{"key":"atypb30","doi-asserted-by":"crossref","unstructured":"P. C. Hansen, J. G. Nagy, and D. P. O'leary,\n                      Deblurring Images: Matrices, Spectra, and Filtering\n                      , SIAM, Philadelphia, 2006,https:\/\/doi.org\/10.1137\/1.9780898718874.","DOI":"10.1137\/1.9780898718874"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1364\/JOSAA.10.001046"},{"key":"atypb32","doi-asserted-by":"crossref","unstructured":"C. Heil,\n                      A Basis Theory Primer: Expanded Edition\n                      , Appl. Numer. Harmon. Anal., Springer Science+ Business Media, New York, 2010,https:\/\/doi.org\/10.1007\/978-0-8176-4687-5.","DOI":"10.1007\/978-0-8176-4687-5"},{"key":"atypb33","unstructured":"B. Hosseini, C. Mougenot, S. Pichardo, E. Constanciel, J. M. Drake, and J. M. Stockie,\n                      A Bayesian Approach for Energy-Based Estimation of Acoustic Aberrations in High Intensity Focused Ultrasound Treatment\n                      , preprint, 2016,https:\/\/arxiv.org\/abs\/1602.08080."},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1137\/16M1076824"},{"key":"atypb35","unstructured":"N. L. Johnson, S. Kotz, and N. Balakrishnan,\n                      Continuous Univariate Distributions, Volume 1\n                      , 2nd ed., John Wiley & Sons, New York, 1994."},{"key":"atypb36","doi-asserted-by":"crossref","unstructured":"J. Kaipio and E. Somersalo,\n                      Statistical and Computational Inverse Problems\n                      , Appl. Math. Sci. 160, Springer Science$+$Business Media, New York, 2005,https:\/\/doi.org\/10.1007\/b138659.","DOI":"10.1007\/b138659"},{"key":"atypb37","doi-asserted-by":"crossref","unstructured":"S. Kotz, T. J. Kozubowski, and K. Podgorski,\n                      The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance\n                      , Springer Science$+$ Business Media, New York, 2012,https:\/\/doi.org\/10.1007\/978-1-4612-0173-1.","DOI":"10.1007\/978-1-4612-0173-1"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1137\/1115038"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/20\/5\/013"},{"key":"atypb40","doi-asserted-by":"crossref","unstructured":"G. Leoni,\n                      A First Course in Sobolev Spaces\n                      , Grad. Stud. Math. 105, American Mathematical Society, Providence, RI, 2009,http:\/\/dx.doi.org\/10.1090\/gsm\/105.","DOI":"10.1090\/gsm\/105\/10"},{"key":"atypb41","unstructured":"W. Linde,\n                      Probability in Banach Spaces: Stable and Infinitely Divisible Distributions\n                      , John Wiley & Sons, New York, 1986."},{"key":"atypb42","unstructured":"F. Lucka,\n                      Bayesian Inversion in Biomedical Imaging\n                      , Ph.D. thesis, University of Mu\u0308nster, Mu\u0308nster, Germany, 2014; available online fromhttp:\/\/nbn-resolving.de\/urn:nbn:de:hbz:6-80359613770."},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1080\/0233188031000078060"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1080\/02664760500079464"},{"key":"atypb45","unstructured":"C. J. Paciorek,\n                      Nonstationary Gaussian Processes for Regression and Spatial Modelling\n                      , Ph.D. thesis, Carnegie Mellon University, Pittsburgh, PA, 2003; available online fromhttp:\/\/www.stat.berkeley.edu\/ paciorek\/diss\/paciorek-thesis.pdf."},{"key":"atypb46","doi-asserted-by":"crossref","unstructured":"S. Peszat and J. Zabczyk,\n                      Stochastic Partial Differential Equations with Le\u0301vy Noise: An Evolution Equation Approach\n                      , Encyclopedia Math. Appl. 113, Cambridge University Press, Cambridge, UK, 2007.","DOI":"10.1017\/CBO9780511721373"},{"key":"atypb47","doi-asserted-by":"crossref","unstructured":"N. G. Polson and J. G. Scott,\n                      Shrink globally, act locally: Sparse Bayesian regularization and prediction\n                      , in Bayesian Statistics 9 (Benidorm, 2010), Oxford University Press, Oxford, UK, 2011, pp. 501-538,https:\/\/doi.org\/10.1093\/acprof:oso\/9780199694587.003.0017.","DOI":"10.1093\/acprof:oso\/9780199694587.003.0017"},{"key":"atypb48","doi-asserted-by":"crossref","unstructured":"C. E. Rasmussen and C. K. I. Williams,\n                      Gaussian Processes for Machine Learning\n                      , MIT Press, Cambridge, MA, 2006.","DOI":"10.7551\/mitpress\/3206.001.0001"},{"key":"atypb49","unstructured":"K. Sato,\n                      Le\u0301vy Processes and Infinitely Divisible Distributions\n                      , Cambridge Stud. Adv. Math. 68, Cambridge University Press, Cambridge, UK, 1999."},{"key":"atypb50","doi-asserted-by":"crossref","unstructured":"F. W. Steutel and K. van Harn,\n                      Infinite Divisibility of Probability Distributions on the Real Line\n                      , Monogr. Textbooks Pure Appl. Math. 259, Marcel Dekker, New York, 2004,https:\/\/doi.org\/10.1201\/9780203014127.","DOI":"10.1201\/9780203014127"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492910000061"},{"key":"atypb52","unstructured":"T. J. Sullivan,\n                      Well-Posed Bayesian Inverse Problems and Heavy-Tailed Stable Banach Space Priors\n                      , preprint, 2016,https:\/\/arxiv.org\/abs\/1605.05898."},{"key":"atypb53","doi-asserted-by":"crossref","unstructured":"M. E. Taylor,\n                      Partial Differential Equations I. Basic Theory\n                      , Appl. Math. Sci. 115, 2nd ed., Springer Science+Business Media, New York, 2011,https:\/\/doi.org\/10.1007\/978-1-4419-7055-8.","DOI":"10.1007\/978-1-4419-7055-8"},{"key":"atypb54","doi-asserted-by":"crossref","unstructured":"M. Unser and P. Tafti,\n                      An Introduction to Sparse Stochastic Processes\n                      , Cambridge University Press, Cambridge, UK, 2013.","DOI":"10.1017\/CBO9781107415805"},{"key":"atypb55","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2014.2311903"},{"key":"atypb56","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2014.2298453"},{"key":"atypb57","doi-asserted-by":"crossref","unstructured":"C. R. Vogel,\n                      Computational Methods for Inverse Problems\n                      , SIAM, Philadelphia, 2002,https:\/\/doi.org\/10.1137\/1.9780898717570.","DOI":"10.1137\/1.9780898717570"},{"key":"atypb58","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/32\/7\/075006"}],"container-title":["SIAM\/ASA Journal on Uncertainty Quantification"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1096372","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:27:24Z","timestamp":1787336844000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1096372"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":58,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/16M1096372"],"URL":"https:\/\/doi.org\/10.1137\/16m1096372","relation":{},"ISSN":["2166-2525"],"issn-type":[{"value":"2166-2525","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}