Don't Forget Its Variance! The Minimum Path Variance Principle for Accurate and Stable Score-Based Density Ratio Estimation

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-DRE (\textbf{Min}imum \textbf{P}ath \textbf{V}ariance DRE), which directly minimizes this missing objective. 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.