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 $\epsilon$. To begin with, we prove that the $\epsilon$-logarithm score is proper (the expected score is optimized when the predictive distribution meets the ground truth) based on discrete approximations. Then, we characterize the probabilistic predictability of an SDS by the optimal expected score and approximate it with an error of scale $\mathcal{O}(\epsilon)$. The approximation quantitatively shows how the system predictability is jointly determined by the neighborhood radius, the differential entropies of process noises, and the system dimension. 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 will converge to the probabilistic predictability 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, we apply the predictability analysis to design unpredictable SDSs. Numerical examples are given to elaborate the results.

翻译：为评估随机动力系统概率预测的质量，评分规则依据预测分布与实测状态分配数值分数。本文提出一种ε-对数评分，通过考虑半径为ε的邻域，推广了经典的对数评分。首先，基于离散近似证明ε-对数评分是恰当的（当预测分布与真实分布一致时期望分数达到最优）。其次，通过最优期望分数刻画随机动力系统的概率可预测性，并以O(ε)量级的误差进行近似。该近似定量揭示了系统可预测性如何由邻域半径、过程噪声的微分熵及系统维度共同决定。除期望分数外，本文还分析了单条轨迹上分数的渐近行为。具体而言，当过程噪声独立同分布时，证明轨迹上的分数将以概率收敛至概率可预测性，且收敛速度关于轨迹长度T为O(T^{-1/2})量级。最后，将该可预测性分析应用于不可预测随机动力系统的设计，并通过数值算例阐述相关结论。