Abstract
Entropy is a measure for the uninformativeness or randomness of a data set, i.e., the higher the entropy is, the lower is the amount of information. In the field of propositional logic it has proven to constitute a suitable measure to be maximized when dealing with models of probabilistic propositional theories. More specifically, it was shown that the model of a probabilistic propositional theory with maximal entropy allows for the deduction of other formulae which are somehow expected by humans, i.e., allows for some kind of common sense reasoning.
In order to pull the technique of maximum entropy entailment to the field of Formal Concept Analysis, we define the notion of entropy of a formal context with respect to the frequency of its object intents, and then define maximum entropy entailment for quantified implication sets, i.e., for sets of partial implications where each implication has an assigned degree of confidence. Furthermore, then this entailment technique is utilized to define so-called maximum entropy implicational bases (ME-bases), and a first general example of such a ME-base is provided.
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Notes
- 1.
In the field of machine learning, an implication is also called association rule, a premise is called antedecent, and a conclusion is called consequent.
- 2.
Of course, this may be easily solved by regarding i as a partial function.
- 3.
We use the logarithm with base 2 here, since we are dealing with data sets or informations, respectively, which are encoded as bits. However, using another base would not cause any problem, since this would only distort the entropy by a multiplicative factor.
- 4.
We denote by
the quantified implication set which assigns no probability to \(X\rightarrow Y\), but otherwise coincides with \(\mathcal {L}\).
the quantified implication set which assigns no probability to References
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The author gratefully thanks the anonymous reviewers for their constructive hints and helpful remarks.
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Kriegel, F. (2017). First Notes on Maximum Entropy Entailment for Quantified Implications. In: Bertet, K., Borchmann, D., Cellier, P., Ferré, S. (eds) Formal Concept Analysis. ICFCA 2017. Lecture Notes in Computer Science(), vol 10308. Springer, Cham. https://doi.org/10.1007/978-3-319-59271-8_10
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DOI: https://doi.org/10.1007/978-3-319-59271-8_10
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