We introduce a new class of real-valued monotones in preordered spaces, injective monotones. We show that the class of preorders for which they exist lies in between the class of preorders with strict monotones and preorders with countable multi-utilities, improving upon the known classification of preordered spaces through real-valued monotones. We extend several well-known results for strict monotones (Richter–Peleg functions) to injective monotones, we provide a construction of injective monotones from countable multi-utilities, and relate injective monotones to classic results concerning (...) Debreu denseness and order separability. Along the way, we connect our results to Shannon entropy and the uncertainty preorder, obtaining new insights into how they are related. In particular, we show how injective monotones can be used to generalize some appealing properties of Jaynes’ maximum entropy principle, which is considered a basis for statistical inference and serves as a justification for many regularization techniques that appear throughout machine learning and decision theory. (shrink)

The study of complexity and optimization in decision theory involves both partial and complete characterizations of preferences over decision spaces in terms of real-valued monotones. With this motivation, and following the recent introduction of new classes of monotones, like injective monotones or strict monotone multi-utilities, we present the classification of preordered spaces in terms of both the existence and cardinality of real-valued monotones and the cardinality of the quotient space. In particular, we take advantage of a characterization of real-valued monotones (...) in terms of separating families of increasing sets to obtain a more complete classification consisting of classes that are strictly different from each other. As a result, we gain new insight into both complexity and optimization, and clarify their interplay in preordered spaces. (shrink)

Legal documents, like contracts or laws, are subject to interpretation. Different people can have different interpretations of the very same document. Large parts of judicial branches all over the world are concerned with settling disagreements that arise, in part, from these different interpretations. In this context, it only seems natural that during the annotation of legal machine learning data sets, disagreement, how to report it, and how to handle it should play an important role. This article presents an analysis of (...) the current state-of-the-art in the annotation of legal machine learning data sets. The results of the analysis show that all of the analysed data sets remove all traces of disagreement, instead of trying to utilise the information that might be contained in conflicting annotations. Additionally, the publications introducing the data sets often do provide little information about the process that derives the “gold standard” from the initial annotations, often making it difficult to judge the reliability of the annotation process. Based on the state-of-the-art, the article provides easily implementable suggestions on how to improve the handling and reporting of disagreement in the annotation of legal machine learning data sets. (shrink)