Active learning for continuous regression has lacked an acquisition function that targets epistemic uncertainty when the predictive distribution is multimodal: variance misses modal disagreement, and information-theoretic targets like BALD are designed for discrete outputs. We introduce a Two-Index framework that makes this separation explicit: one stochastic index selects among competing model hypotheses (epistemic source), while a second governs within-hypothesis randomness (aleatoric source). An entropy decomposition within the framework identifies the mutual information between the output and the epistemic index as a principled acquisition objective, and we prove this quantity vanishes as the model is trained on growing datasets, confirming that it captures exactly the uncertainty data can resolve. Because this mutual information is intractable for continuous outputs, we derive the Mutual Information Lower Bound (MI-LB) acquisition function, a closed-form approximation for Mixture Density Network ensembles. On benchmarks featuring multimodal systems, MI-LB matches or beats every baseline evaluated and is the only method to do so consistently -- geometric and Fisher-based baselines compete only when the input space already encodes the multimodality, and collapse otherwise.

翻译：连续回归的主动学习在预测分布呈多模态时，一直缺乏针对认知不确定性的采集函数：方差会忽略模态分歧，而BALD等基于信息论的指标是为离散输出设计的。我们提出了一种双指数框架，使这种分离变得明确：一个随机指数从多个竞争模型假设中进行选择（认知源），而第二个指数则控制假设内部的随机性（偶然源）。该框架内的熵分解将输出与认知指数之间的互信息识别为一种原则性的采集目标，并且我们证明，随着模型在日益增大的数据集上训练，该量会消失，证实它恰好捕获了数据能够解决的不确定性。由于这种互信息对于连续输出而言是难以处理的，我们推导了互信息下界（MI-LB）采集函数，它是混合密度网络集成的闭式近似。在以多模态系统为特征的基准测试中，MI-LB匹配或超越了所有被评估的基线方法，并且是唯一始终如一做到这一点的方法——几何基线和Fisher基线仅在输入空间已编码了多模态性时才能与之竞争，否则就会失效。