AI 中文总结
本研究针对混合分布关注量，提出混合变分近似的变分GO-OED方法，解决后验分布含原子的下界估计问题，在船舶横摇安全评估中实现稳定的EIG下界估计并识别出有价值的海浪条件。
AI 中文摘要
目标导向最优实验设计（GO-OED）依据关注量（QoI）的期望信息增益（EIG）而非完整参数向量来选择实验。本研究针对概率力学中出现的混合离散-连续QoI分布，开发了一种变分GO-OED公式；这类分布由阈值化或基于事件的变换产生，将具有正概率的不确定输入集合映射为共同值，而其他输入则产生连续变化的响应。研究的激励应用是随机海浪中的船舶横摇安全评估，其中QoI为超过规定横摇角阈值的时间超越概率：当未发生超越时该量为零，否则在正值范围内连续变化。纯连续变分近似无法主导包含原子的后验QoI分布，会产生无限的Kullback-Leibler散度和Barber-Agakov下界的平凡值-∞；用连续密度值对原子样本评分会改变目标，无法生成有效的下界估计器。我们引入一种混合变分近似，分别对条件原子概率和连续分量建模，其中连续分量采用归一化流（normalizing flow）。解析示例可恢复正确的EIG分布，而船舶横摇应用则提供了稳定的EIG下界估计，并识别出用于时间超越概率推断的有价值海浪条件。
英文摘要
Goal-oriented optimal experimental design (GO-OED) selects experiments according to the expected information gain (EIG) about a quantity of interest (QoI) rather than the full parameter vector. This work develops a variational GO-OED formulation for mixed discrete-continuous QoI laws arising in probabilistic mechanics when thresholding or event-based transformations map a positive-probability set of uncertain inputs to a common value while other inputs produce continuously varying responses. The motivating application is ship roll safety assessment in random waves, where the QoI is the temporal exceedance probability above a prescribed roll-angle threshold. This quantity is zero when no exceedance occurs and varies continuously over positive values otherwise. A purely continuous variational approximation does not dominate a posterior QoI law containing an atom, yielding an infinite Kullback-Leibler divergence and a trivial Barber-Agakov lower bound of $-\infty$. Scoring atom samples using continuous density values instead changes the objective and does not produce a valid lower-bound estimator. We introduce a mixed variational approximation that models the conditional atom probability and continuous component separately, with a normalizing flow used for the latter. An analytical example recovers the correct EIG landscape, while the ship roll application provides stable EIG lower-bound estimates and identifies informative wave conditions for temporal-exceedance-probability inference.
