通过有限探针构建漂移模型的分布可信度评估,可量化误差并自动拒绝不可靠结论。
Finite-Probe Total-Variation Certificates for Finite-Basis Drifting Models

- 基于有限位置的向量场观测,推导出分布差异的后验总变差上界。
- 在高斯RBF模型下实现无需分布假设的统计保证,且支持自适应误差控制。
- 适用于需要严格可信度验证的密度估计场景,如科学模拟与安全关键系统。
漂移目标通过在有限位置观测到的带噪声向量场比较目标分布与模型分布。我们研究这种固定测量系统能得出何种分布结论。对于可积反对称相互作用和指定有限密度基下的绝对连续分布,未归一化采样分子满足 \operatorname{vec}(V_X)=Mc,其中 $c$ 为反对称失配,$M$ 依赖于探针。该恒等式导出一个后验总变差(TV)置信上界,考虑了保留场噪声、估计算子误差以及归一化密度近似在张成空间中的外部验证 $L^1$ 残差半径;非正可观测性裕度返回平凡上界并拒绝判断。审计通过保留样本重新计算该分子;归一化漂移统计需单独进行联合分子-分母分析。对于高斯-RBF相互作用,全局包络支持分布无关和经验伯努利半径,无需截断,并提供原始漂移目标中拉普拉斯相似性的配套边界。通过总体格矩阵刻画随机探针可观测性,识别秩与对称性退化,并证明大带宽下趋向均值匹配。合成实验验证了高斯与拉普拉斯分子,分别预设有界向量与方差自适应半径,蒙特卡洛校准算子,归一化有限基近似周围非零残差半径,向外圆化可观测性边界及设计拒判。联合基大小/维度应力路径将评估扩展至 $m=8$。结果为有限密度类或具有外部残差半径的归一化有限基密度近似提供条件诊断,而非从小训练漂移获得的通用保证。
原文摘要 · Abstract (English)
Drifting objectives compare a target and model distribution through a vector field observed noisily at finitely many locations. We ask what distributional conclusion such a frozen measurement system warrants. For integrable antisymmetric interactions and absolutely continuous laws in a declared finite density basis, the unnormalized sampled numerator satisfies $\operatorname{vec}(V_X)=Mc$, where $c$ is an antisymmetric mismatch and $M$ is probe-dependent. This identity yields an a posteriori total-variation (TV) upper confidence bound accounting for held-out field noise, estimated-operator error, and externally validated $L^1$ residual radii around normalized density approximants in the span; a nonpositive observability margin returns the trivial TV bound and abstains. The audit recomputes this numerator from held-out samples; a normalized drift statistic requires a separate joint numerator--denominator analysis. For Gaussian-RBF interactions, a global envelope supports distribution-free and empirical-Bernstein radii without truncation, with companion bounds for the Laplace similarity in the original drifting objective. We characterize random-probe observability by a population Gram matrix, identify rank and symmetry degeneracies, and prove large-bandwidth collapse toward mean matching. Synthetic studies exercise Gaussian and Laplace numerators, separately prespecified bounded-vector and variance-adaptive radii, Monte Carlo-calibrated operators, nonzero residual radii around normalized finite-basis approximants, outward-rounded observability bounds, and designed abstention. A joint basis-size/dimension stress path extends evaluation through $m=8$. The result is a conditional diagnostic for a finite density class, or for normalized finite-basis density approximants with external residual radii, not a universal guarantee from small training drift.
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