IMPRECISE PROBABILITIES AND THEIR APPLICATIONS

30 June - 2 July 1999

** Abstract **

Bayesian implicative analysis was proposed for summarizing the association in a $2\times 2$ contingency table in terms possibly, asymmetrical such as, \eg, ``presence of feature $a$ implies, in general, presence of feature $b$'' (``$a$ quasi-implies $b$'' in short). Here, we consider the multivariate version of this problem: having $n$ units which are classified according to $\Qcard$ binary questions, we want to summarize the association between questions in terms of quasi-implications between features. We will first show how at a descriptive level the notion of implication can be weakened into that of quasi-implication. The inductive step assumes that the $n$ units are a sample from a $2^\Qcard$-multinomial population. Uncertainty about the patterns' true frequencies is expressed by an imprecise Dirichlet model which yields upper and lower posterior probabilities for any quasi-implicative statement. This model is shown to have several advantages over the Bayesian models based on a single Dirichlet prior, especially whenever $2^\Qcard$ is large and many patterns are thus unobserved by design.

** Keywords. ** Quasi-implication, logical model, measure of association, multivariate implicative index, Boolean methods, Bayesian inference, upper and lower probabilities.

The paper is available in the following formats and sites:

- Gzipped postscript file (Granada - Spain)
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- Gzipped postscript file (Gent - Belgium)
- Postscript file (Granada - Spain)
- Postscript file (Sao Paulo - Brasil)
- Postscript file (Gent - Belgium)
- Gzipped pdf file (Granada - Spain)
- pdf file (Granada - Spain)
- pdf file (Sau Paulo - Brasil)
- pdf file (Gent - Belgium)

Laboratoire Cognition et Activités Finalisées

Université Paris 8 & CNRS ESA 7021

2 rue de la Liberté

93526 Saint-Denis Cedex 2, France

** E-mail addresses: **

Jean-Marc Bernard | berj@univ-paris8.fr |

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