AI summaryⓘ
The authors study how well one can compare two probability distributions using noisy measurements of a vector field at limited points, called drifting objectives. They develop mathematical tools to provide upper confidence bounds on the difference between distributions, accounting for measurement noise and estimation errors. Their approach applies to certain smooth interactions like Gaussian and Laplace kernels, and they analyze when the problem is well-observed or inherently ambiguous. Through experiments with synthetic data, they demonstrate how their diagnostic method can identify when the comparison is reliable or should be abstained from. Overall, the work provides a way to judge the quality of distribution comparison in finite-data, noisy settings rather than giving universal guarantees.
drifting objectivesvector fieldtotal variation distanceGaussian-RBF kernelLaplace kernelobservabilityfinite density basisconfidence boundsMonte Carlo calibrationnormalized density approximants
Abstract
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.