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This short note describes and proves a connectedness property which was introduced in Blocher et al. [2023] in the context of data depth functions for partial orders. The connectedness property gives a structural insight into union-free generic sets. These sets, presented in Blocher et al. [2023], are defined by using a closure operator on the set of all partial orders which naturally appears within the theory of formal concept analysis. In the language of formal concept analysis, the property of connectedness can be vividly proven. However, since within Blocher et al. [2023] we did not discuss formal concept analysis, we outsourced the proof to this note.

Spaces with locally varying scale of measurement, like multidimensional structures with differently scaled dimensions, are pretty common in statistics and machine learning. Nevertheless, it is still understood as an open question how to exploit the entire information encoded in them properly. We address this problem by considering an order based on (sets of) expectations of random variables mapping into such non-standard spaces. This order contains stochastic dominance and expectation order as extreme cases when no, or respectively perfect, cardinal structure is given. We derive a (regularized) statistical test for our proposed generalized stochastic dominance (GSD) order, operationalize it by linear optimization, and robustify it by imprecise probability models. Our findings are illustrated with data from multidimensional poverty measurement, finance, and medicine., Comment: Accepted for the 39th Conference on Uncertainty in Artificial Intelligence (UAI 2023)

We propose a framework for descriptively analyzing sets of partial orders based on the concept of depth functions. Despite intensive studies of depth functions in linear and metric spaces, there is very little discussion on depth functions for non-standard data types such as partial orders. We introduce an adaptation of the well-known simplicial depth to the set of all partial orders, the union-free generic (ufg) depth. Moreover, we utilize our ufg depth for a comparison of machine learning algorithms based on multidimensional performance measures. Concretely, we analyze the distribution of different classifier performances over a sample of standard benchmark data sets. Our results promisingly demonstrate that our approach differs substantially from existing benchmarking approaches and, therefore, adds a new perspective to the vivid debate on the comparison of classifiers., Comment: Accepted to ISIPTA 2023; Forthcoming in: Proceedings of Machine Learning Research