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However, for more challenging systems a substantial error can be present even after many iterations have been performed. The estimates obtained in this case are of little value unless further information can be provided about the numerical error. In this paper we propose a novel statistical model for this numerical error set in a Bayesian framework. Our approach is a strict generalisation of the conjugate gradient method, which is recovered as the posterior mean for a particular choice of prior. The estimates obtained are analysed with Krylov subspace methods and a contraction result for the posterior is presented. The method is then analysed in a simulation study as well as being applied to a challenging problem in medical imaging.","url_abs":"https://arxiv.org/abs/1801.05242v3","url_pdf":"https://arxiv.org/pdf/1801.05242v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-bayesian-conjugate-gradient-method","repo_url":"https://github.com/jcockayne/bcg","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"unanswered"}},{"paper_slug":"a-bayesian-conjugate-gradient-method","repo_url":"https://github.com/AndreySavinov/BayessianCG","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null},{"paper_slug":"a-bayesian-conjugate-gradient-method","repo_url":"https://github.com/VLSF/BayesKrylov","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1801.05242","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1801.05242"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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