Concepts of a Discrete Random Variable

Abstract

A formal concept is defined in the literature as a pair (extent, intent) with respect to a context which is usually empirical, as for example a sample of transactions. This is somewhat unsatisfying since concepts, though born from experiences, should not depend on them. In this paper we consider the above concepts as ‘empirical concepts’ and we define the notion of concept, in a context-free framework, as a limit intent, by proving, applying the large number law, that : Given a random variable χ taking its value in a countable σ-semilattice, the random intents of empirical concepts, with respect to a sample of χ, converge almost everywhere to a fixed deterministic limit, called a concept, whose identification shows that it only depends on the distribution Pχ of χ. Moreover, the set of such concepts is the σ-semilattice generated by the support of χ and has even a structure of σ-lattice: the lattice concept of a random variable.We also compute the mean number of concepts and frequent itemsets for a hierarchical Bernoulli mixtures model. Last, we propose an algorithm to find out maximal frequent itemsets by using minimal winning coalitions of Pχ.KeywordsFrequent ItemsetsGalois TheoryWinning CoalitionDiscrete Random VariableGalois ConnectionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.