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Proximal nested sampling for high-dimensional Bayesian model selection

Proximal nested sampling for high-dimensional Bayesian model selection
Proximal nested sampling for high-dimensional Bayesian model selection

Bayesian model selection provides a powerful framework for objectively comparing models directly from observed data, without reference to ground truth data. However, Bayesian model selection requires the computation of the marginal likelihood (model evidence), which is computationally challenging, prohibiting its use in many high-dimensional Bayesian inverse problems. With Bayesian imaging applications in mind, in this work we present the proximal nested sampling methodology to objectively compare alternative Bayesian imaging models for applications that use images to inform decisions under uncertainty. The methodology is based on nested sampling, a Monte Carlo approach specialised for model comparison, and exploits proximal Markov chain Monte Carlo techniques to scale efficiently to large problems and to tackle models that are log-concave and not necessarily smooth (e.g., involving ℓ 1 or total-variation priors). The proposed approach can be applied computationally to problems of dimension O(10 6) and beyond, making it suitable for high-dimensional inverse imaging problems. It is validated on large Gaussian models, for which the likelihood is available analytically, and subsequently illustrated on a range of imaging problems where it is used to analyse different choices of dictionary and measurement model.

Bayesian evidence, Inverse problems, MCMC sampling, Marginal likelihood, Model selection, Nested sampling, Proximal optimisation
0960-3174
Cai, Xiaohao
de483445-45e9-4b21-a4e8-b0427fc72cee
McEwen, Jason D.
6346d637-52cd-4533-a411-bd657d45ccc2
Pereyra, Marcelo
7ae249d9-94ea-4f67-a3ec-e2907665952e
Cai, Xiaohao
de483445-45e9-4b21-a4e8-b0427fc72cee
McEwen, Jason D.
6346d637-52cd-4533-a411-bd657d45ccc2
Pereyra, Marcelo
7ae249d9-94ea-4f67-a3ec-e2907665952e

Cai, Xiaohao, McEwen, Jason D. and Pereyra, Marcelo (2022) Proximal nested sampling for high-dimensional Bayesian model selection. Statistics and Computing, 32 (5), [87]. (doi:10.1007/s11222-022-10152-9).

Record type: Article

Abstract

Bayesian model selection provides a powerful framework for objectively comparing models directly from observed data, without reference to ground truth data. However, Bayesian model selection requires the computation of the marginal likelihood (model evidence), which is computationally challenging, prohibiting its use in many high-dimensional Bayesian inverse problems. With Bayesian imaging applications in mind, in this work we present the proximal nested sampling methodology to objectively compare alternative Bayesian imaging models for applications that use images to inform decisions under uncertainty. The methodology is based on nested sampling, a Monte Carlo approach specialised for model comparison, and exploits proximal Markov chain Monte Carlo techniques to scale efficiently to large problems and to tackle models that are log-concave and not necessarily smooth (e.g., involving ℓ 1 or total-variation priors). The proposed approach can be applied computationally to problems of dimension O(10 6) and beyond, making it suitable for high-dimensional inverse imaging problems. It is validated on large Gaussian models, for which the likelihood is available analytically, and subsequently illustrated on a range of imaging problems where it is used to analyse different choices of dictionary and measurement model.

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Accepted/In Press date: 9 September 2022
Published date: 5 October 2022
Additional Information: Funding Information: This work was supported by the Leverhulme Trust and by EPSRC grants EP/T007346/1 and EP/W007673/1. The authors would like to thank the editor and two anonymous reviewers for their valuable suggestions to improve the manuscript. The authors are also grateful to Abdul-Lateef Haji-Ali for helpful comments. For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising. Publisher Copyright: © 2022, The Author(s).
Keywords: Bayesian evidence, Inverse problems, MCMC sampling, Marginal likelihood, Model selection, Nested sampling, Proximal optimisation

Identifiers

Local EPrints ID: 471550
URI: http://eprints.soton.ac.uk/id/eprint/471550
ISSN: 0960-3174
PURE UUID: e86a3d05-ccc2-410d-a720-556cc002ef15
ORCID for Xiaohao Cai: ORCID iD orcid.org/0000-0003-0924-2834

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Date deposited: 10 Nov 2022 17:44
Last modified: 12 Nov 2022 02:56

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Contributors

Author: Xiaohao Cai ORCID iD
Author: Jason D. McEwen
Author: Marcelo Pereyra

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