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However, traditional approximate inference techniques make little to no use of this available information. We propose the framework of\n                    <jats:italic>post-process Bayesian inference<\/jats:italic>\n                    as a means to obtain a quick posterior approximation from existing target density evaluations, with no further model calls. Within this framework, we introduce Variational Sparse Bayesian Quadrature (\n                    <jats:sc>vsbq<\/jats:sc>\n                    ), a method for post-process approximate inference for models with\n                    <jats:italic>black-box<\/jats:italic>\n                    and potentially noisy likelihoods.\n                    <jats:sc>vsbq<\/jats:sc>\n                    reuses existing target density evaluations to build a sparse Gaussian process (GP) surrogate model of the log posterior density function. Subsequently, we leverage sparse-GP Bayesian quadrature combined with variational inference to achieve fast approximate posterior inference over the surrogate. We validate our method on challenging synthetic scenarios and real-world applications from computational neuroscience. The experiments show that\n                    <jats:sc>vsbq<\/jats:sc>\n                    builds high-quality posterior approximations by post-processing existing optimization traces, with no further model evaluations.\n                  <\/jats:p>","DOI":"10.1007\/s11222-025-10695-7","type":"journal-article","created":{"date-parts":[[2025,8,11]],"date-time":"2025-08-11T02:50:40Z","timestamp":1754880640000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fast post-process Bayesian inference with Variational Sparse Bayesian Quadrature"],"prefix":"10.1007","volume":"35","author":[{"given":"Chengkun","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gr\u00e9goire","family":"Clart\u00e9","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martin","family":"J\u00f8rgensen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luigi","family":"Acerbi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,8,11]]},"reference":[{"key":"10695_CR1","first-page":"8222","volume":"31","author":"L Acerbi","year":"2018","unstructured":"Acerbi, L.: Variational Bayesian Monte Carlo. 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