Implicative analysis for multivariate binary data using an imprecise Dirichlet model

Abstract

Bayesian implicative analysis was proposed for summarizing the association in a 2×2 contingency table in possibly asymmetrical terms such as “presence of feature a implies, usually, 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 q 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 q -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 when 2 q is large and many patterns are thus unobserved by design.