Admissible Generalizations of Examples as Rules Rule learning is a data analysis task consisting ofextracting rules to generalize examples. For a data scientist, some generalizations called here admissible generalizations, make more sense. We explore formal properties of admissible generalizations.A formalization for generalization of examples is proposed that allows to express rule admissibility. Some admissible generalizations are captured by topological operators. We examine selecting supersets of examples that induce these operators and we define classes of such choice functions. This formalization is particularly developed in the case of numerical attributes. Classes of such functions are associated with notions of generalization and they are used to comment some behaviours of the CN2 algorithm.

Generalized transitivity: A systematic comparison of concepts with an application to preferences in the Babington Smith model Abstract   Reciprocal relations are binary relations Q with entries    Q  (  i  ,  j  )  xe2x88x88    0  ,  1     , and such that    Q  (  i  ,  j  )    Q  (  j  ,  i  )    1   . Relations of this kind occur quite naturally in various domains, such as preference modeling and preference learning. For example,      q    i  ,  j      could be the fraction of voters in a population who prefer candidate i to candidate j. In the literature, various attempts have been made at generalizing the notion of transitivity to reciprocal relations. In this paper, we compare three important frameworks of generalized transitivity: gstochastic transitivity, Ttransitivity, and cycletransitivity. To this end, we introduce Etransitivity as an even more general notion. We also use this framework to extend an existing hierarchy of different types of transitivity. As an illustration, we study transitivity properties of probabilities of pairwise preferences, which are induced as marginals of an underlying probability distribution on rankings (strict total orders) of a set of alternatives. In particular, we analyze the interesting case of the socalled Babington Smith model, a parametric family of distributions of that kind.