  Hierarchical multi-label classification (HMC) has drawn increasing attention
in the past few decades. It is applicable when hierarchical relationships among
classes are available and need to be incorporated along with the multi-label
classification whereby each object is assigned to one or more classes. There
are two key challenges in HMC: i) optimizing the classification accuracy, and
meanwhile ii) ensuring the given class hierarchy. To address these challenges,
in this article, we introduce a new statistic called the multidimensional local
precision rate (mLPR) for each object in each class. We show that
classification decisions made by simply sorting objects across classes in
descending order of their true mLPRs can, in theory, ensure the class hierarchy
and lead to the maximization of CATCH, an objective function we introduce that
is related to the area under a hit curve. This approach is the first of its
kind that handles both challenges in one objective function without additional
constraints, thanks to the desirable statistical properties of CATCH and mLPR.
In practice, however, true mLPRs are not available. In response, we introduce
HierRank, a new algorithm that maximizes an empirical version of CATCH using
estimated mLPRs while respecting the hierarchy. The performance of this
approach was evaluated on a synthetic data set and two real data sets; ours was
found to be superior to several comparison methods on evaluation criteria based
on metrics such as precision, recall, and $F_1$ score.
