Score-based methods have emerged as a powerful framework for density ratio estimation (DRE), but they face an important paradox in that, while theoretically path-independent, their practical performance depends critically on the chosen path schedule. We resolve this issue by proving that tractable training objectives differ from the ideal, ground-truth objective by a crucial, overlooked term: the path variance of the time score. To address this, we propose MinPV (\textbf{Min}imum \textbf{P}ath \textbf{V}ariance) Principle, which introduces a principled heuristic to minimize the overlooked path variance. Our key contribution is the derivation of a closed-form expression for the variance, turning an intractable problem into a tractable optimization. By parameterizing the path with a flexible Kumaraswamy Mixture Model, our method learns a data-adaptive, low-variance path without heuristic selection. This principled optimization of the complete objective yields more accurate and stable estimators, establishing new state-of-the-art results on challenging benchmarks.

翻译：基于分数匹配的方法已成为密度比估计（DRE）的强大框架，但其面临一个重要悖论：尽管理论上具有路径无关性，其实际性能却严重依赖于所选的路径调度。我们通过证明可处理的训练目标与理想的真实目标之间存在一个关键且被忽视的项——时间分数的路径方差——来解决此问题。为此，我们提出最小路径方差（MinPV）原理，该原理引入了一种原则性启发式方法以最小化被忽视的路径方差。我们的核心贡献是推导出了该方差的闭式表达式，从而将一个难以处理的问题转化为可处理的优化问题。通过使用灵活的Kumaraswamy混合模型对路径进行参数化，我们的方法能够学习到一种数据自适应的低方差路径，而无需启发式选择。这种对完整目标进行的原则性优化产生了更准确、更稳定的估计器，并在具有挑战性的基准测试中取得了新的最先进结果。