# 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.

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