To assess the quality of a probabilistic prediction for stochastic dynamical systems (SDSs), scoring rules assign a numerical score based on the predictive distribution and the measured state. In this paper, we propose an $\epsilon$-logarithm score that generalizes the celebrated logarithm score by considering a neighborhood with radius $\epsilon$. We characterize the probabilistic predictability of an SDS by optimizing the expected score over the space of probability measures. We show how the probabilistic predictability is quantitatively determined by the neighborhood radius, the differential entropies of process noises, and the system dimension. Given any predictor, we provide approximations for the expected score with an error of scale $\mathcal{O}(\epsilon)$. In addition to the expected score, we also analyze the asymptotic behaviors of the score on individual trajectories. Specifically, we prove that the score on a trajectory can converge to the expected score when the process noises are independent and identically distributed. Moreover, the convergence speed against the trajectory length $T$ is of scale $\mathcal{O}(T^{-\frac{1}{2}})$ in the sense of probability. Finally, numerical examples are given to elaborate the results.

翻译：为评估随机动力系统（SDS）概率预测的质量，评分规则根据预测分布和实测状态给出数值评分。本文提出一种$\epsilon$-对数评分，该评分通过考虑半径为$\epsilon$的邻域推广了经典的对数评分。我们通过在概率测度空间上优化期望评分来刻画随机动力系统的概率可预测性。我们证明了概率可预测性如何由邻域半径、过程噪声的微分熵以及系统维度定量决定。对于任意预测器，我们以$\mathcal{O}(\epsilon)$量级的误差给出了期望评分的近似表达式。除期望评分外，我们还分析了评分在个体轨迹上的渐近行为。具体而言，我们证明了当过程噪声独立同分布时，轨迹上的评分可收敛至期望评分。此外，以概率意义衡量，其关于轨迹长度$T$的收敛速度为$\mathcal{O}(T^{-\frac{1}{2}})$量级。最后，通过数值算例对结果进行了阐释。