Probabilistic implication bases in FCA and probabilistic bases of GCIs in

Abstract

A probabilistic formal context is a triadic context the third dimension of which is a set of worlds equipped with a probability measure. After a formal definition of this notion, this document introduces probability of implications with respect to probabilistic formal contexts, and provides a construction for a base of implications the probabilities of which exceed a given lower threshold. A comparison between confidence and probability of implications is drawn, which yields the fact that both measures do not coincide. Furthermore, the results are extended towards the lightweight description logic $\mathcal{E}{\mathcal{L}}^{\perp }$ with probabilistic interpretations, and a method for computing a base of general concept inclusions the probabilities of which are greater than a pre-defined lower bound is proposed. Additionally, we consider so-called probabilistic attributes over probabilistic formal contexts, and provide a method for the axiomatization of implications over probabilistic attributes.

Keywords:

• Formal Concept Analysis
• probabilistic formal context
• implication
• description logics
• probabilistic interpretation
• general concept inclusion
• probabilistic attribute

Acknowledgements

The author thanks Sebastian Rudolph for proof reading and a fruitful discussion, and furthermore the anonymous reviewers for their constructive comments.

Notes

No potential conflict of interest was reported by the author.

1 For two intents X and Y, we say that X is a lower intent-neighbor of Y if $X⊊Y$ and there is no intent between X and Y. Furthermore, we denote this as $X\prec Y$.