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标题: "Posterior contraction rates in Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families"
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## Mathematics > Statistics Theory

## Title:Posterior contraction rates in Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families

Authors:[Emanuele Dolera](https://arxiv.org/search/math?searchtype=author&query=Dolera,+E), [Stefano Favaro](https://arxiv.org/search/math?searchtype=author&query=Favaro,+S), [Matteo Giordano](https://arxiv.org/search/math?searchtype=author&query=Giordano,+M)

[View PDF](https://arxiv.org/pdf/2608.11130) [HTML (experimental)](https://arxiv.org/html/2608.11130v1)

> Abstract:We study posterior contraction in positive-order Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families. We embed the natural parameter in a Hilbert scale and model it via a standard Gaussian series prior expanded in the eigenbasis generating the scale. Under a two-sided link condition on the Fisher information and suitable local regularity assumptions, we show that smoothness-matching priors achieve minimax-optimal posterior contraction rates in any Hilbert scale norm up to the regularity of the ground truth. Our analysis builds on the novel approach to posterior contraction based on the Wasserstein distance recently introduced by Dolera et al. (2024). It combines refined Laplace-type estimates for infinite-dimensional integrals associated to the posterior kernels with a mixed-geometry estimate controlling their stability under fluctuations in the data, itself resting on a tailored Poincaré inequality for posterior distributions conditioned on neighbourhoods of the truth. We apply the general theory to density estimation with a logistic parametrisation, Poisson intensity estimation with an exponential link, and the Gaussian white-noise model, yielding minimax contraction rates in Sobolev norms across all three settings. In particular, these yield optimal recovery of density score functions and derivatives of Poisson intensities.

| Comments: |  |
| --- | --- |
| Subjects: | Statistics Theory (math.ST); Machine Learning (stat.ML) |
| Cite as: | [arXiv:2608.11130](https://arxiv.org/abs/2608.11130) \[math.ST\] |
|  | (or [arXiv:2608.11130v1](https://arxiv.org/abs/2608.11130v1) \[math.ST\] for this version) |
|  | [https://doi.org/10.48550/arXiv.2608.11130](https://doi.org/10.48550/arXiv.2608.11130) |

## Submission history

From: Matteo Giordano \[[view email](https://arxiv.org/show-email/dbc0edda/2608.11130)\]  
**\[v1\]** Tue, 11 Aug 2026 16:45:55 UTC (54 KB)

[Which authors of this paper are endorsers?](https://arxiv.org/auth/show-endorsers/2608.11130) | Disable MathJax ([What is MathJax?](https://info.arxiv.org/help/mathjax.html))
