The Boltzmann-Shannon Index (BSI) for clustered continuous data is introduced as a normalized measure that captures the relationship between geometry-based and frequency-based probability distributions defined over the clusters. In essence, it quantifies the similarity across densities of the clusters, which are defined by a given labeling. This labeling may originate from a geometric partitioning of the state space itself, but need not in general. We illustrate its performance on synthetic Gaussian mixtures, the Iris benchmark data set, and a high-imbalance resource-allocation scenario, showing that the BSI provides a coherent assessment in cases where traditional metrics give incomplete or misleading signals. Moreover, in the resource-allocation setting where equal density may be associated with a "fair" distribution, we demonstrate that BSI not only detects inequality with high sensitivity, but also offers a numerically smooth measure that can be easily embedded in optimization frameworks as a regularization term for modern policy-making. Finally, the BSI also offers a new measure of the effectiveness for a given symbolic representation, i.e. coarse-grain states, for continuous-valued data recorded from complex dynamical systems.

翻译：本文提出的玻尔兹曼-香农指数（BSI）是针对聚类连续数据的归一化度量，它捕捉了基于几何与基于频率的概率分布在聚类上的关系。本质上，该指数量化了由给定标签定义的各聚类密度之间的相似性。此类标签可能源于状态空间本身的几何划分，但通常并非必需。我们通过合成高斯混合模型、Iris基准数据集以及高不平衡资源分配场景展示了其性能，表明BSI在传统指标给出不完整或误导性信号的情况下能提供一致的评估。此外，在资源分配场景中，等密度可能关联于“公平”分布，我们证明BSI不仅能以高灵敏度检测不平等性，还提供了一种数值平滑的度量，可轻松作为正则化项嵌入优化框架，用于现代政策制定。最后，BSI还为复杂动力系统记录的连续值数据的给定符号表示（即粗粒度状态）提供了一种新的有效性度量。