Discussion of “Frequentist coverage of adaptive nonparametric Bayesian credible sets”
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VI & VIIFirst I would like to congratulate the three authors for a very nice paper.During a visit to Eindhoven in 2010, Botond Szabó and Harry van Zanten mentioned the first steps of this work to me, which concerned the understanding of certain empirical Bayes procedures in the white noise L 2 -setting.Since then, together with Aad van der Vaart, they have broadened their original goals and have produced an impressive and very interesting series of papers on the subject.The present paper is indeed one aspect of a larger body of work, and we will mention a few connections with these related papers below.The authors start from the signal in white noise model, that after projection in L 2 onto an appropriate basis, typically related to the SVD of the operator K of the inverse problem, is translated into a sequence formulation.They choose a prior distribution that makes coordinates independent:1. Priors for Bayesian credible sets.2. Bayesian credible regions and simulations.
估计纵向因果剂量反应关系的贝叶斯框架
Existing causal methods for time-varying exposure and time-varying confounding focus on estimating the average causal effect of a time-varying binary treatment on an end-of-study outcome, offering limited tools for characterizing marginal causal dose-response relationships under continuous exposures. We propose a scalable, nonparametric Bayesian framework for estimating marginal longitudinal causal dose-response functions with repeated outcome measurements. Our approach…
RBF 核高斯过程最大似然估计的精确渐近理论
Gaussian processes (GPs) are widely used across machine learning, spatial statistics, time-series analysis, optimization, Bayesian statistics, and scientific applications. A central component of a GP model is its kernel, which is typically specified through a parametric family. Among the most widely used choices is the radial basis function (RBF), also known as the squared exponential or Gaussian kernel, owing to its simple form, smoothness, and flexibility. In practice, the…
贝叶斯算子学习:高斯先验下后验存在性与点估计收敛
We develop a Bayesian framework for learning nonlinear operators between infinite-dimensional spaces. Given a map $g_0:\mathcal{X}\to\mathcal{Y}$ between separable Hilbert spaces, we study the recovery of $g_0$ from $n\in\mathbb{N}$ noisy input-output pairs $(\boldsymbol{X},\boldsymbol{Z})=(X_i,Z_i)_{i=1}^n$ with $Z_i= g_0 (X_i ) + E_i$. Here the $X_i\in\mathcal{X}$ are randomly drawn 'design' points in a compact subset of $\mathcal X$, and the $E_i$ are assumed to be i.i.d.…
面向高维数据的残差树高斯过程建模框架
With the advance of measurement technologies and increasing computing power, large spatial data with heterogeneous structures are often collected over high-dimensional domains. Existing Gaussian process (GP) models and computational strategies are often inadequate for analyzing such datasets in multi-dimensional domains. To address these challenges, we develop a Bayesian residual tree GP methodology called ResTGP for large spatial data with potentially heterogeneous…
二维 Navier–Stokes 方程的数据同化:后验测度的最优高斯渐近性质
A functional Bernstein–von Mises theorem is proved for posterior measures arising in a data assimilation problem with the two-dimensional Navier–Stokes equation where a Gaussian process prior is assigned to the initial condition of the system. The posterior measure, which provides the update in the space of all trajectories arising from a discrete sample of the (deterministic) dynamics, is shown to be approximated by a Gaussian random vector field obtained as the solution to…
处理是否有益?有序结局因果推断的贝叶斯方法
Randomized experiments with ordinal outcomes are common across scientific applications, but conventional causal estimands such as the average treatment effect are difficult to interpret because ordinal categories lack meaningful numerical spacing. Two interpretable estimands, the probabilities that the treatment is beneficial and strictly beneficial, depend on the joint distribution of the potential outcomes and are therefore not identifiable from observed data. The sharp…