Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems
In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020. https://doi.org/10.1007/s00211-020-01131-1 ), where it is shown that in such a settin...
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| Veröffentlicht in: | Numerische Mathematik 2022-02, Vol.150 (2), p.521-549 |
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| creator | Helin, Tapio Kretschmann, Remo |
| description | In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020.
https://doi.org/10.1007/s00211-020-01131-1
), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties. |
| doi_str_mv | 10.1007/s00211-021-01266-9 |
| format | Article |
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https://doi.org/10.1007/s00211-020-01131-1
), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties.</description><identifier>ISSN: 0029-599X</identifier><identifier>EISSN: 0945-3245</identifier><identifier>DOI: 10.1007/s00211-021-01266-9</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Approximation ; Bayesian analysis ; Errors ; Estimates ; Inverse problems ; Mapping ; Mathematical analysis ; Mathematical and Computational Engineering ; Mathematical and Computational Physics ; Mathematical Methods in Physics ; Mathematics ; Mathematics and Statistics ; Noise levels ; Nonlinearity ; Numerical Analysis ; Numerical and Computational Physics ; Perturbation ; Simulation ; Statistical inference ; Theoretical</subject><ispartof>Numerische Mathematik, 2022-02, Vol.150 (2), p.521-549</ispartof><rights>The Author(s) 2022</rights><rights>The Author(s) 2022. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c363t-dd2a43103a4f734ca0d9ba8e9467f084dc32d6477dff50c2748f43943edc40b93</citedby><cites>FETCH-LOGICAL-c363t-dd2a43103a4f734ca0d9ba8e9467f084dc32d6477dff50c2748f43943edc40b93</cites><orcidid>0000-0002-5788-0008 ; 0000-0002-4856-955X</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00211-021-01266-9$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00211-021-01266-9$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>315,781,785,27928,27929,41492,42561,51323</link.rule.ids></links><search><creatorcontrib>Helin, Tapio</creatorcontrib><creatorcontrib>Kretschmann, Remo</creatorcontrib><title>Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems</title><title>Numerische Mathematik</title><addtitle>Numer. Math</addtitle><description>In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020.
https://doi.org/10.1007/s00211-020-01131-1
), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties.</description><subject>Approximation</subject><subject>Bayesian analysis</subject><subject>Errors</subject><subject>Estimates</subject><subject>Inverse problems</subject><subject>Mapping</subject><subject>Mathematical analysis</subject><subject>Mathematical and Computational Engineering</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Methods in Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Noise levels</subject><subject>Nonlinearity</subject><subject>Numerical Analysis</subject><subject>Numerical and Computational Physics</subject><subject>Perturbation</subject><subject>Simulation</subject><subject>Statistical inference</subject><subject>Theoretical</subject><issn>0029-599X</issn><issn>0945-3245</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp9UMtOAzEMjBBIlMIPcIrEOeA8NmmOUPGSKriABKcozQO2aneXZIvo35NlkbhxsceyZ2wPQqcUzimAusgAjFJSAgHKpCR6D01Ai4pwJqr9goFpUmn9coiOcl4BUCUFnaDXh7YhNu82Xd_2tcMhpTbhkPt6Y_uQcSxV_x7wwnZr6wK2XZfar6FZtw2uG3xldyHXdsCfIeWAS3-5Dpt8jA6iXedw8pun6Pnm-ml-RxaPt_fzywVxXPKeeM-s4BS4FVFx4Sx4vbSzoIVUEWbCO868FEr5GCtwTIlZFFwLHrwTsNR8is5G3bL4Y1suN6t2m5qy0jDJFFVKFPUpYuOUS23OKUTTpfJF2hkKZrDQjBaaEsyPhWaQ5iMpl-HmLaQ_6X9Y39bEdP0</recordid><startdate>20220201</startdate><enddate>20220201</enddate><creator>Helin, Tapio</creator><creator>Kretschmann, Remo</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-5788-0008</orcidid><orcidid>https://orcid.org/0000-0002-4856-955X</orcidid></search><sort><creationdate>20220201</creationdate><title>Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems</title><author>Helin, Tapio ; Kretschmann, Remo</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c363t-dd2a43103a4f734ca0d9ba8e9467f084dc32d6477dff50c2748f43943edc40b93</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Approximation</topic><topic>Bayesian analysis</topic><topic>Errors</topic><topic>Estimates</topic><topic>Inverse problems</topic><topic>Mapping</topic><topic>Mathematical analysis</topic><topic>Mathematical and Computational Engineering</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Methods in Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Noise levels</topic><topic>Nonlinearity</topic><topic>Numerical Analysis</topic><topic>Numerical and Computational Physics</topic><topic>Perturbation</topic><topic>Simulation</topic><topic>Statistical inference</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Helin, Tapio</creatorcontrib><creatorcontrib>Kretschmann, Remo</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><jtitle>Numerische Mathematik</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Helin, Tapio</au><au>Kretschmann, Remo</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems</atitle><jtitle>Numerische Mathematik</jtitle><stitle>Numer. Math</stitle><date>2022-02-01</date><risdate>2022</risdate><volume>150</volume><issue>2</issue><spage>521</spage><epage>549</epage><pages>521-549</pages><issn>0029-599X</issn><eissn>0945-3245</eissn><abstract>In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020.
https://doi.org/10.1007/s00211-020-01131-1
), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00211-021-01266-9</doi><tpages>29</tpages><orcidid>https://orcid.org/0000-0002-5788-0008</orcidid><orcidid>https://orcid.org/0000-0002-4856-955X</orcidid><oa>free_for_read</oa></addata></record> |
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| subjects | Approximation Bayesian analysis Errors Estimates Inverse problems Mapping Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Noise levels Nonlinearity Numerical Analysis Numerical and Computational Physics Perturbation Simulation Statistical inference Theoretical |
| title | Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems |
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