发表机构
George Mason University（乔治梅森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对高斯过程变量选择中ARD和导数灵敏度缺乏校准规则的问题，提出学生化乘子自助法校准导数过程灵敏度，实现FDR控制，减少虚假输入，成本低。
AI 中文摘要
自动相关性确定（ARD）是高斯过程（GP）回归中变量选择的默认工具，它通过逆长度尺度对输入进行排序——这些尺度衡量函数变化的速度，而非输入对预测的贡献程度——并且没有提供校准的规则来决定保留哪些输入。以预测为中心的替代方案，即导数灵敏度 $\nu_j = \mathbb{E}[(\partial f/\partial x_j)^2]$，可以从拟合的GP中以封闭形式获得，但将其转化为选择规则比看起来更难：在零输入处，估计量是退化的二次型，因此Wald和Bernstein-von Mises截断是反保守的，而自然的残差自助法（residual bootstrap）的尺度是错误的。我们证明，对GP导数过程进行学生化乘子自助法（studentized multiplier bootstrap）可以修复这两个问题，并通过二次型的不变性原理证明其有效性，并在输入间获得渐近的族wise误差和错误发现率控制。在100次重复中，该规则在输入确实为零时控制了FDR，而未校准的导数排名违反目标高达2倍，Bernstein-von Mises截断违反2.2倍；在匹配的FDR下，它不损失功效；在Matérn核和输入相关性高达0.99的情况下依然成立；在具有植入和真实零输入的真实数据上，它减少了5-12倍的虚假输入；其成本为GP拟合的5-18%；并且块平均变体在成本随$n$线性增长的情况下保持有效性。
英文摘要
Automatic relevance determination (ARD), the default tool for variable selection in Gaussian-process (GP) regression, ranks inputs by inverse lengthscales -- which measure how fast a function varies, not how much an input contributes to prediction -- and offers no calibrated rule for deciding which inputs to keep. The prediction-centred alternative, the derivative sensitivity $ν_j = \mathbb{E}[(\partial f/\partial x_j)^2]$, is available in closed form from a fitted GP, but turning it into a selection rule is harder than it looks: at a null input the estimator is a degenerate quadratic form, so Wald and Bernstein-von Mises cutoffs are anti-conservative, and the natural residual bootstrap is mis-scaled. We show that a studentized multiplier bootstrap of the GP derivative process repairs both, prove its validity through an invariance principle for quadratic forms, and obtain asymptotic family-wise and false-discovery-rate control across inputs. Over 100 replications the rule controls FDR wherever inputs are truly null, while uncalibrated derivative rankings breach the target by up to 2x and a Bernstein-von Mises cutoff by 2.2x; at matched FDR it loses no power; it holds under a Matérn kernel and input correlation up to 0.99; on real data with planted and authentic null inputs it admits 5-12x fewer spurious inputs; it costs 5-18% of the GP fit; and a block-averaged variant retains validity at cost linear in $n$.
