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FIRST INTERNATIONAL
SYMPOSIUM ON

IMPRECISE PROBABILITIES AND THEIR APPLICATIONS

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Ghent, Belgium

30 June - 2 July 1999

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** ELECTRONIC PROCEEDINGS **

## Jean-Marc Bernard

# Implicative Analysis for Multivariate Binary Data using an Imprecise Dirichlet Model

** 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:

** Authors addresses: **

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: **

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