无穷维指数族的Sobolev范数下的后验收缩率与贝叶斯导数估计
Posterior contraction rates in Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families
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中文总结 AI 辅助
该研究针对无穷维指数族，基于Wasserstein距离的新方法，证明匹配光滑度的先验可实现Sobolev范数下的极小极大最优后验收缩率，并在三类场景中验证了该理论的有效性。
中文摘要 AI 辅助
我们研究无穷维指数族在正阶Sobolev范数下的后验收缩及贝叶斯导数估计问题。我们将自然参数嵌入希尔伯特尺度，通过在生成该尺度的特征基上展开的标准高斯级数先验对其建模。在Fisher信息的双侧连接条件和合适的局部正则性假设下，我们证明匹配光滑度的先验能达到任意希尔伯特尺度范数下的极小极大最优后验收缩率，该收缩率上限为真实值的正则性。我们的分析基于Dolera等人（2024）新近提出的基于Wasserstein距离的后验收缩新方法，将与后验核相关的无穷维积分的精细拉普拉斯型估计，与控制其在数据波动下稳定性的混合几何估计相结合，而混合几何估计本身依赖于针对真实值邻域条件下后验分布的定制庞加莱不等式。我们将该通用理论应用于带逻辑参数化的密度估计、带指数连接的泊松强度估计及高斯白噪声模型，在这三种场景下均得到Sobolev范数下的极小极大收缩率，尤其可实现密度得分函数和泊松强度导数的最优恢复。
英文摘要
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 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, Probab. Theory Relat. Fields). It combines Laplace-type approximations 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 under the logistic parametrisation, Poisson intensity estimation under the exponential link, and mildly ill-posed linear inverse problems observed in Gaussian white noise. The resulting minimax rates yield optimal recovery of density score functions, derivatives of Poisson intensities and, in a concrete elliptic inverse problem, the unknown source function.
发表机构
- University of Pavia（帕维亚大学）
- University of Turin（都灵大学）
- Collegio Carlo Alberto, Turin（都灵卡洛阿尔贝托学院）
机构由 AI 辅助整理，请以论文原文为准。
