# MAP estimators and their consistency in Bayesian nonparametric inverse problems

- Handle URI:
- http://hdl.handle.net/10754/594280
- Title:
- MAP estimators and their consistency in Bayesian nonparametric inverse problems
- Authors:
- Abstract:
- We consider the inverse problem of estimating an unknown function u from noisy measurements y of a known, possibly nonlinear, map applied to u. We adopt a Bayesian approach to the problem and work in a setting where the prior measure is specified as a Gaussian random field μ0. We work under a natural set of conditions on the likelihood which implies the existence of a well-posed posterior measure, μy. Under these conditions, we show that the maximum a posteriori (MAP) estimator is well defined as the minimizer of an Onsager-Machlup functional defined on the Cameron-Martin space of the prior; thus, we link a problem in probability with a problem in the calculus of variations. We then consider the case where the observational noise vanishes and establish a form of Bayesian posterior consistency for the MAP estimator. We also prove a similar result for the case where the observation of can be repeated as many times as desired with independent identically distributed noise. The theory is illustrated with examples from an inverse problem for the Navier-Stokes equation, motivated by problems arising in weather forecasting, and from the theory of conditioned diffusions, motivated by problems arising in molecular dynamics. © 2013 IOP Publishing Ltd.
- KAUST Department:
- Citation:
- Dashti M, Law KJH, Stuart AM, Voss J (2013) MAP estimators and their consistency in Bayesian nonparametric inverse problems. Inverse Problems 29: 095017. Available: http://dx.doi.org/10.1088/0266-5611/29/9/095017.
- Publisher:
- Journal:
- Issue Date:
- 1-Sep-2013
- DOI:
- 10.1088/0266-5611/29/9/095017
- ARXIV:
- arXiv:1303.4795v3
- Type:
- Article
- ISSN:
- 0266-5611; 1361-6420

- Appears in Collections:
- Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

# Full metadata record

DC Field | Value | Language |
---|---|---|

dc.contributor.author | Dashti, M. | en |

dc.contributor.author | Law, K. J H | en |

dc.contributor.author | Stuart, A. M. | en |

dc.contributor.author | Voss, J. | en |

dc.date.accessioned | 2016-01-19T14:45:05Z | en |

dc.date.available | 2016-01-19T14:45:05Z | en |

dc.date.issued | 2013-09-01 | en |

dc.identifier.citation | Dashti M, Law KJH, Stuart AM, Voss J (2013) MAP estimators and their consistency in Bayesian nonparametric inverse problems. Inverse Problems 29: 095017. Available: http://dx.doi.org/10.1088/0266-5611/29/9/095017. | en |

dc.identifier.issn | 0266-5611 | en |

dc.identifier.issn | 1361-6420 | en |

dc.identifier.doi | 10.1088/0266-5611/29/9/095017 | en |

dc.identifier.uri | http://hdl.handle.net/10754/594280 | en |

dc.description.abstract | We consider the inverse problem of estimating an unknown function u from noisy measurements y of a known, possibly nonlinear, map applied to u. We adopt a Bayesian approach to the problem and work in a setting where the prior measure is specified as a Gaussian random field μ0. We work under a natural set of conditions on the likelihood which implies the existence of a well-posed posterior measure, μy. Under these conditions, we show that the maximum a posteriori (MAP) estimator is well defined as the minimizer of an Onsager-Machlup functional defined on the Cameron-Martin space of the prior; thus, we link a problem in probability with a problem in the calculus of variations. We then consider the case where the observational noise vanishes and establish a form of Bayesian posterior consistency for the MAP estimator. We also prove a similar result for the case where the observation of can be repeated as many times as desired with independent identically distributed noise. The theory is illustrated with examples from an inverse problem for the Navier-Stokes equation, motivated by problems arising in weather forecasting, and from the theory of conditioned diffusions, motivated by problems arising in molecular dynamics. © 2013 IOP Publishing Ltd. | en |

dc.publisher | IOP Publishing | en |

dc.title | MAP estimators and their consistency in Bayesian nonparametric inverse problems | en |

dc.type | Article | en |

dc.contributor.department | Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division | en |

dc.identifier.journal | Inverse Problems | en |

dc.contributor.institution | Department of Mathematics, University of Sussex, Brighton BN1 9QH, United Kingdom | en |

dc.contributor.institution | Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom | en |

dc.contributor.institution | School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom | en |

dc.identifier.arxivid | arXiv:1303.4795v3 | en |

kaust.author | Law, Kody | en |

All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.