Quantifying distributional separation across groups is fundamental in statistical learning and scientific discovery, yet most classical discrepancy measures are tailored to two-group comparisons. We generalize the underlap coefficient (UNL), a multi-group separation measure, to multivariate variables. We establish key properties of UNL and provide an explicit connection to the total variation. We further interpret the UNL as a dependence measure between a group label and variables of interest and compare it with mutual information. We propose an importance sampling estimator of the UNL that can be combined with flexible density estimators. The utility of the UNL for assessing partition-covariate dependence in clustering is highlighted in detail, where it is particularly useful for evaluating the single-weights assumption in covariate-dependent mixture models. Finally we illustrate the application of the UNL in clustering using two real world datasets.

翻译：量化不同组别间的分布分离程度是统计学习和科学发现的基础，然而大多数经典差异度量方法仅适用于双组比较。本文将重叠系数（UNL）——一种多组分离度量——推广至多元变量。我们建立了UNL的关键性质，并给出了其与总变差的显式联系。进一步将UNL解释为组别标签与目标变量间的依赖度量，并与互信息进行比较。我们提出了一种可与灵活密度估计器结合的重要性抽样估计量用于计算UNL。文中重点阐述了UNL在评估聚类中划分-协变量依赖关系的实用价值，该指标特别适用于检验协变量依赖混合模型中的单权重假设。最后，我们通过两个真实数据集展示了UNL在聚类分析中的应用。