Proximal nested sampling for high-dimensional Bayesian model selection

TL;DR

This paper introduces proximal nested sampling, a scalable Monte Carlo method for Bayesian model selection in high-dimensional imaging problems, capable of handling complex priors and large datasets.

Contribution

It develops a novel proximal nested sampling approach that efficiently computes model evidence for high-dimensional Bayesian inverse problems with non-smooth priors.

Findings

Successfully applied to Gaussian models with analytical likelihoods

Demonstrated scalability to problems of dimension over 10^6

Enabled comparison of different imaging models and priors

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.