Bayesian inverse problems with Monte Carlo forward models
The full application of Bayesian inference to inverse problems requires exploration of a posterior distribution that typically does not possess a standard form. In this context, Markov chain Monte Carlo (MCMC) methods are often used. These methods require many evaluations of a computationally intens...
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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American Institute of Mathematical Sciences (AIMS)
2013
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| Online Access: | http://hdl.handle.net/1721.1/78669 https://orcid.org/0000-0001-8242-3290 |
| _version_ | 1826199581432479744 |
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| author | Bal, Guillaume Langmore, Ian Marzouk, Youssef M. |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Bal, Guillaume Langmore, Ian Marzouk, Youssef M. |
| author_sort | Bal, Guillaume |
| collection | MIT |
| description | The full application of Bayesian inference to inverse problems requires exploration of a posterior distribution that typically does not possess a standard form. In this context, Markov chain Monte Carlo (MCMC) methods are often used. These methods require many evaluations of a computationally intensive forward model to produce the equivalent of one independent sample from the posterior. We consider applications in which approximate forward models at multiple resolution levels are available, each endowed with a probabilistic error estimate. These situations occur, for example, when the forward model involves Monte Carlo integration. We present a novel MCMC method called MC[superscript 3] that uses low-resolution forward models to approximate draws from a posterior distribution built with the high-resolution forward model. The acceptance ratio is estimated with some statistical error; then a confidence interval for the true acceptance ratio is found, and acceptance is performed correctly with some confidence. The high-resolution models are rarely run and a significant speed up is achieved.
Our multiple-resolution forward models themselves are built around a new importance sampling scheme that allows Monte Carlo forward models to be used efficiently in inverse problems. The method is used to solve an inverse transport problem that finds applications in atmospheric remote sensing. We present a path-recycling methodology to efficiently vary parameters in the transport equation. The forward transport equation is solved by a Monte Carlo method that is amenable to the use of MC[superscript 3] to solve the inverse transport problem using a Bayesian formalism. |
| first_indexed | 2024-09-23T11:22:32Z |
| format | Article |
| id | mit-1721.1/78669 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T11:22:32Z |
| publishDate | 2013 |
| publisher | American Institute of Mathematical Sciences (AIMS) |
| record_format | dspace |
| spelling | mit-1721.1/786692022-10-01T03:09:21Z Bayesian inverse problems with Monte Carlo forward models Bal, Guillaume Langmore, Ian Marzouk, Youssef M. Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Marzouk, Youssef M. The full application of Bayesian inference to inverse problems requires exploration of a posterior distribution that typically does not possess a standard form. In this context, Markov chain Monte Carlo (MCMC) methods are often used. These methods require many evaluations of a computationally intensive forward model to produce the equivalent of one independent sample from the posterior. We consider applications in which approximate forward models at multiple resolution levels are available, each endowed with a probabilistic error estimate. These situations occur, for example, when the forward model involves Monte Carlo integration. We present a novel MCMC method called MC[superscript 3] that uses low-resolution forward models to approximate draws from a posterior distribution built with the high-resolution forward model. The acceptance ratio is estimated with some statistical error; then a confidence interval for the true acceptance ratio is found, and acceptance is performed correctly with some confidence. The high-resolution models are rarely run and a significant speed up is achieved. Our multiple-resolution forward models themselves are built around a new importance sampling scheme that allows Monte Carlo forward models to be used efficiently in inverse problems. The method is used to solve an inverse transport problem that finds applications in atmospheric remote sensing. We present a path-recycling methodology to efficiently vary parameters in the transport equation. The forward transport equation is solved by a Monte Carlo method that is amenable to the use of MC[superscript 3] to solve the inverse transport problem using a Bayesian formalism. United States. Dept. of Energy (Early Career Research Program Grant DE-SC0003908) 2013-05-02T14:49:31Z 2013-05-02T14:49:31Z 2013-02 2012-12 Article http://purl.org/eprint/type/JournalArticle 1930-8337 http://hdl.handle.net/1721.1/78669 Marzouk, Youssef, Ian Langmore, and Guillaume Bal. “Bayesian Inverse Problems with Monte Carlo Forward Models.” Inverse Problems and Imaging 7.1 (2013): 81–105. ©2013 America Institute of Mathematical Sciences https://orcid.org/0000-0001-8242-3290 en_US http://dx.doi.org/10.3934/ipi.2013.7.81 Inverse Problems and Imaging Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf American Institute of Mathematical Sciences (AIMS) American Institute of Mathematical Sciences |
| spellingShingle | Bal, Guillaume Langmore, Ian Marzouk, Youssef M. Bayesian inverse problems with Monte Carlo forward models |
| title | Bayesian inverse problems with Monte Carlo forward models |
| title_full | Bayesian inverse problems with Monte Carlo forward models |
| title_fullStr | Bayesian inverse problems with Monte Carlo forward models |
| title_full_unstemmed | Bayesian inverse problems with Monte Carlo forward models |
| title_short | Bayesian inverse problems with Monte Carlo forward models |
| title_sort | bayesian inverse problems with monte carlo forward models |
| url | http://hdl.handle.net/1721.1/78669 https://orcid.org/0000-0001-8242-3290 |
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