函数型数据均值的贝叶斯单样本检验
A Bayesian One-Sample Test for the Mean of Functional Data
浏览论文内容
中文总结 AI 辅助
针对函数型数据均值是否为零的检验问题，提出基于RKHS有限近似与尖峰-平板先验的贝叶斯方法，采用RJMCMC估计后验，并验证了收缩率与实证效果。
中文摘要 AI 辅助
在函数型数据分析的背景下，我们提出了一种贝叶斯检验，用于评估高斯总体均值函数是否恒为零。通过利用与底层过程协方差函数相关联的再生核希尔伯特空间（RKHS）的结构，我们将均值近似为核截面的有限线性组合。这种有限维表示源于均值位于相应RKHS中的自然假设，从而能够通过系数上的尖峰-平板先验实现参数化贝叶斯假设检验方法。鉴于无限维空间上缺乏规范参考测度，似然函数通过诱导高斯测度之间的Radon-Nikodym导数来表示。由于核截面的数量未知，我们采用可逆跳跃马尔可夫链蒙特卡洛（RJMCMC）方法来探索跨维参数空间并估计后验分布。从所得的后验样本中，我们构建了原假设后验概率和贝叶斯因子的估计量，为反对原假设提供了可量化的证据，并支持在不确定性下做出明智决策。此外，为验证该方法，我们在温和条件下为所提出的检验建立了近最优的后验收缩率，在模拟数据集上展示了其经验性能，并利用真实气候数据说明了其应用。
英文摘要
In the context of functional data analysis, we propose a Bayesian test to assess whether the mean function of a Gaussian population is identically zero. By leveraging the structure of the reproducing kernel Hilbert space (RKHS) associated with the covariance function of the underlying process, we approximate the mean by a finite linear combination of kernel sections. This finite-dimensional representation arises from the natural assumption that the mean lies in the corresponding RKHS, enabling a parametric Bayesian approach to hypothesis testing through a spike-and-slab prior on the coefficients. Given the absence of a canonical reference measure on infinite-dimensional spaces, the likelihood is expressed via a Radon-Nikodym derivative between induced Gaussian measures. Since the number of kernel sections is unknown, we employ reversible jump Markov chain Monte Carlo (RJMCMC) to explore the trans-dimensional parameter space and estimate the posterior distribution. From the resulting posterior samples we construct estimators of the posterior probability of the null hypothesis and the Bayes factor, providing quantifiable evidence against the null and supporting informed decisions under uncertainty. Furthermore, to validate this methodology we establish near-optimal posterior contraction rates for the proposed test under mild conditions, demonstrate its empirical performance on simulated data sets, and illustrate its application with real climate data.
发表机构
- Universidad Autónoma de Madrid（马德里自治大学）
机构由 AI 辅助整理，请以论文原文为准。
