{"id":2073,"date":"2025-02-26T07:03:54","date_gmt":"2025-02-26T07:03:54","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/02\/26\/2502-18279\/"},"modified":"2025-02-26T07:03:54","modified_gmt":"2025-02-26T07:03:54","slug":"2502-18279","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/02\/26\/2502-18279\/","title":{"rendered":"Near-Optimal Approximations for Bayesian Inference in Function Space"},"content":{"rendered":"

Near-Optimal Approximations for Bayesian Inference in Function Space
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arXiv:2502.18279v1 Announce Type: new
\nAbstract: We propose a scalable inference algorithm for Bayes posteriors defined on a reproducing kernel Hilbert space (RKHS). Given a likelihood function and a Gaussian random element representing the prior, the corresponding Bayes posterior measure $Pi_{text{B}}$ can be obtained as the stationary distribution of an RKHS-valued Langevin diffusion. We approximate the infinite-dimensional Langevin diffusion via a projection onto the first $M$ components of the Kosambi-Karhunen-Lo`eve expansion. Exploiting the thus obtained approximate posterior for these $M$ components, we perform inference for $Pi_{text{B}}$ by relying on the law of total probability and a sufficiency assumption. The resulting method scales as $O(M^3+JM^2)$, where $J$ is the number of samples produced from the posterior measure $Pi_{text{B}}$. Interestingly, the algorithm recovers the posterior arising from the sparse variational Gaussian process (SVGP) (see Titsias, 2009) as a special case, owed to the fact that the sufficiency assumption underlies both methods. However, whereas the SVGP is parametrically constrained to be a Gaussian process, our method is based on a non-parametric variational family $mathcal{P}(mathbb{R}^M)$ consisting of all probability measures on $mathbb{R}^M$. As a result, our method is provably close to the optimal $M$-dimensional variational approximation of the Bayes posterior $Pi_{text{B}}$ in $mathcal{P}(mathbb{R}^M)$ for convex and Lipschitz continuous negative log likelihoods, and coincides with SVGP for the special case of a Gaussian error likelihood.<\/div>\n

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\n Veit Wild, James Wu, Dino Sejdinovic, Jeremias Knoblauch
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Near-Optimal Approximations for Bayesian Inference in Function Space arXiv:2502.18279v1 Announce Type: new Abstract: We propose a scalable inference algorithm for Bayes posteriors defined on a reproducing kernel Hilbert space (RKHS). Given a likelihood function and a Gaussian random element representing the prior, the corresponding Bayes posterior measure $Pi_{text{B}}$ can be obtained as the stationary distribution […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,113,183,112],"tags":[338,193,1863],"class_list":["post-2073","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-lg","category-stat-me","category-stat-ml","tag-gaussian","tag-inference","tag-posterior"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/2073"}],"collection":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/comments?post=2073"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/2073\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=2073"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=2073"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=2073"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}