Multivariate Bayesian implicative analysis for a binary questionnaire: quasi-implications and simplified Galois lattice

Abstract

We propose a new method for simplifying the Galois lattice associated to a binary questionnaire (n units classified according to q binary questions). The method consists in weakening the implications borne by the lattice into quasi-implications. At the descriptive level, the method involves a new measure for quasi-implications (the multivariate implicative index) which satisfies some requirements of invariance by logical equivalence. At the inductive level, uncertainty about the patterns' true frequencies is expressed by an imprecise-Dirichlet model. This model is shown to have several advantages over the usual non-informative Bayesian approach based on a single Dirichlet prior, especially for the case where n is small in comparison to 2q. An important feature of the method is that it provides two implicative summaries, descriptive and inductive, which both constitute simplified versions of the initial Galois lattice.