### Abstract

In this paper, we propose a general method to derive an upper bound for the contraction rate of the posterior distribution for nonparametric inverse problems. We present a general theorem that allows us to derive contraction rates for the parameter of interest from contraction rates of the related direct problem of estimating transformed parameter of interest. An interesting aspect of this approach is that it allows us to derive contraction rates for priors that are not related to the singular value decomposition of the operator. We apply our result to several examples of linear inverse problems, both in the white noise sequence model and the nonparametric regression model, using priors based on the singular value decomposition of the operator, location-mixture priors and splines prior, and recover minimax adaptive contraction rates.

Language | English |
---|---|

Pages | 2091-2121 |

Number of pages | 31 |

Journal | Bernoulli |

Volume | 24 |

Issue number | 3 |

DOIs | |

State | Published - Aug 2018 |

### Fingerprint

### Keywords

- Bayesian nonparametrics
- Modulus of continuity
- Nonparametric inverse problems
- Posterior distribution
- Rate of contraction

### Cite this

*Bernoulli*,

*24*(3), 2091-2121. DOI: 10.3150/16-BEJ921

}

*Bernoulli*, vol. 24, no. 3, pp. 2091-2121. DOI: 10.3150/16-BEJ921

**A general approach to posterior contraction in nonparametric inverse problems.** / Knapik, Bartek; Salomond, Jean Bernard.

Research output: Contribution to Journal › Article › Academic › peer-review

TY - JOUR

T1 - A general approach to posterior contraction in nonparametric inverse problems

AU - Knapik,Bartek

AU - Salomond,Jean Bernard

PY - 2018/8

Y1 - 2018/8

N2 - In this paper, we propose a general method to derive an upper bound for the contraction rate of the posterior distribution for nonparametric inverse problems. We present a general theorem that allows us to derive contraction rates for the parameter of interest from contraction rates of the related direct problem of estimating transformed parameter of interest. An interesting aspect of this approach is that it allows us to derive contraction rates for priors that are not related to the singular value decomposition of the operator. We apply our result to several examples of linear inverse problems, both in the white noise sequence model and the nonparametric regression model, using priors based on the singular value decomposition of the operator, location-mixture priors and splines prior, and recover minimax adaptive contraction rates.

AB - In this paper, we propose a general method to derive an upper bound for the contraction rate of the posterior distribution for nonparametric inverse problems. We present a general theorem that allows us to derive contraction rates for the parameter of interest from contraction rates of the related direct problem of estimating transformed parameter of interest. An interesting aspect of this approach is that it allows us to derive contraction rates for priors that are not related to the singular value decomposition of the operator. We apply our result to several examples of linear inverse problems, both in the white noise sequence model and the nonparametric regression model, using priors based on the singular value decomposition of the operator, location-mixture priors and splines prior, and recover minimax adaptive contraction rates.

KW - Bayesian nonparametrics

KW - Modulus of continuity

KW - Nonparametric inverse problems

KW - Posterior distribution

KW - Rate of contraction

UR - http://www.scopus.com/inward/record.url?scp=85041900413&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85041900413&partnerID=8YFLogxK

U2 - 10.3150/16-BEJ921

DO - 10.3150/16-BEJ921

M3 - Article

VL - 24

SP - 2091

EP - 2121

JO - Bernoulli: A Journal of Mathematical Statistics and Probability

T2 - Bernoulli: A Journal of Mathematical Statistics and Probability

JF - Bernoulli: A Journal of Mathematical Statistics and Probability

SN - 1350-7265

IS - 3

ER -