Near-Optimal Approximations for Bayesian Inference in Function Space
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arXiv
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| Format: | Preprint |
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2025
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| author | Wild, Veit Wu, James Sejdinovic, Dino Knoblauch, Jeremias |
| author_facet | Wild, Veit Wu, James Sejdinovic, Dino Knoblauch, Jeremias |
| contents | 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 $Π_{\text{B}}$ 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-Loève expansion. Exploiting the thus obtained approximate posterior for these $M$ components, we perform inference for $Π_{\text{B}}$ 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 $Π_{\text{B}}$. 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 $Π_{\text{B}}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18279 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Near-Optimal Approximations for Bayesian Inference in Function Space Wild, Veit Wu, James Sejdinovic, Dino Knoblauch, Jeremias Machine Learning Methodology 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 $Π_{\text{B}}$ 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-Loève expansion. Exploiting the thus obtained approximate posterior for these $M$ components, we perform inference for $Π_{\text{B}}$ 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 $Π_{\text{B}}$. 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 $Π_{\text{B}}$ 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. |
| title | Near-Optimal Approximations for Bayesian Inference in Function Space |
| topic | Machine Learning Methodology |
| url | https://arxiv.org/abs/2502.18279 |