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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<jats:italic>proximal nested sampling<\/jats:italic>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<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell _1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>\u2113<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>or total-variation priors). The proposed approach can be applied computationally to problems of dimension<jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {O}}(10^6)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>O<\/mml:mi><mml:mo>(<\/mml:mo><mml:msup><mml:mn>10<\/mml:mn><mml:mn>6<\/mml:mn><\/mml:msup><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>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.<\/jats:p>","DOI":"10.1007\/s11222-022-10152-9","type":"journal-article","created":{"date-parts":[[2022,10,5]],"date-time":"2022-10-05T09:05:02Z","timestamp":1664960702000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Proximal nested sampling for high-dimensional Bayesian model selection"],"prefix":"10.1007","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0924-2834","authenticated-orcid":false,"given":"Xiaohao","family":"Cai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jason D.","family":"McEwen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marcelo","family":"Pereyra","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,10,5]]},"reference":[{"key":"10152_CR1","doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. 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