We introduce and prove a \textbf{Conditional Score Expectation (CSE)} identity: an exact relation for the marginal score of affine diffusion processes that links scores across time via a conditional expectation under the forward dynamics. Motivated by this identity, we propose a CSE-based statistical estimator for the score using a Self-Normalized Importance Sampling (SNIS) procedure with prior samples and forward noise. We analyze its relationship to the standard Tweedie estimator, proving anti-correlation for Gaussian targets and establishing the same behavior for general targets in the small time-step regime. Exploiting this structure, we derive a variance-minimizing blended score estimator given by a state--time dependent convex combination of the CSE and Tweedie estimators. Numerical experiments show that this optimal-blending estimator reduces variance and improves sample quality for a fixed computational budget compared to either baseline. We further extend the framework to Bayesian inverse problems via likelihood-informed SNIS weights, and demonstrate improved reconstruction quality and sample diversity on high-dimensional image reconstruction tasks and PDE-governed inverse problems.

翻译：本文提出并证明了一个**条件得分期望（CSE）恒等式**：该精确关系描述了仿射扩散过程的边缘得分，通过前向动力学下的条件期望将不同时刻的得分联系起来。受此恒等式启发，我们提出一种基于CSE的统计估计器，利用先验样本和前向噪声，通过自归一化重要性采样（SNIS）程序来估计得分。我们分析了该估计器与标准Tweedie估计器的关系，证明了在高斯目标下的反相关性，并确立了在小时间步长条件下一般目标的相同行为。利用此结构，我们推导出一个方差最小化的混合得分估计器，它由CSE与Tweedie估计器随状态和时间变化的凸组合给出。数值实验表明，在固定计算预算下，与任一基线方法相比，此最优混合估计器能降低方差并提升样本质量。我们进一步通过似然信息加权的SNIS权重将该框架扩展至贝叶斯逆问题，并在高维图像重建任务和偏微分方程约束的逆问题上展示了改进的重建质量与样本多样性。