Proximal nested sampling for high-dimensional Bayesian model selection

Xiaohao Cai*, Jason D. McEwen, Marcelo Pereyra

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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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 ℓ1 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.

Original languageEnglish
Article number87
JournalStatistics and Computing
Volume32
Issue number5
DOIs
Publication statusPublished - 5 Oct 2022

Keywords

  • Bayesian evidence
  • Inverse problems
  • Marginal likelihood
  • MCMC sampling
  • Model selection
  • Nested sampling
  • Proximal optimisation

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Computational Theory and Mathematics

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