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Visualizing the behavior and some symmetry properties of Bayesian confirmation measures

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Abstract

Bayesian confirmation measures, a special class of interestingness measures, are functions usually adopted in ranking inductive rules generated by data mining methods such as association rule mining, decision trees, rough sets. Till now a plethora of measures have been defined in many different ways. Identifying and effectively distinguishing among them is a difficult task. In this paper we propose a unified visual approach aimed at comparing and classifying a large subset of Bayesian confirmation measures (those satisfying the initial and final probability dependence condition). We first reduce the set of variables in their analytical expression to only two, thus allowing to draw their contour lines on the plane. We observe that two dimensional contour lines plots represent a sort of fingerprints of the confirmation measures and, therefore, this geometric visualization can be used as an effective tool in order to investigate properties and behavior of the measures. We highlight the potential of this approach not only to study known measures but also in order to invent new measures satisfying given required characteristics. We finally define, following the geometry of the plots, a new set of symmetry properties of confirmation measures and describe geometrically four classical symmetries.

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Notes

  1. IFPD stands for initial and final probability dependence (Crupi et al. 2010)

  2. ROC stands for receiver operating characteristics (see e.g. Fawcett 2006)

  3. Conjunction effects are “situations in which two hypotheses H1 and H2 are both confirmed by some evidence E, and in which the conjunction \(H1\wedge H2\) is even more highly confirmed” (Atkinson 2012).

  4. Even if in the general definition of confirmation space, i.e. the domain of a Bayesian confirmation measure c(x, y), we stated that it corresponds to the open square (0,1)x(0,1) in many cases it can be extended to the square (0,1]x(0,1) by including also the points of the line x=1, i.e. \(P(H{\vert }E)=1\).

  5. Flach (2003) and Fürnkranz and Flach (2003a) discovered rather similar geometric symmetries of evaluation metrics on the ROC space but mentioned them without an explicit definition.

  6. The logarithm has been used in order to meet Bayesian confirmation measures sign condition.

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  1. Department of Management, University Ca’ Foscari, Venezia, Italy

    Emilio Celotto

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  1. Emilio Celotto
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Correspondence to Emilio Celotto.

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Responsible editor: Johannes Fürnkranz.

The present paper is part of the research project Development of a Web marketing tool for assessing and positioning tourist-cultural destinations jointly supported by the Department of Management of the University Ca’ Foscari-Venezia, CISET-International Centre for Studies on Tourism Economics and Fondazione Ca’ Foscari Venezia.

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Celotto, E. Visualizing the behavior and some symmetry properties of Bayesian confirmation measures. Data Min Knowl Disc 31, 739–773 (2017). https://doi.org/10.1007/s10618-016-0487-5

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