This paper is an attempt to set a justi…cation for making use of some dicrepancy indexes, starting from the classical Maximum Likelihood de…nition, and adapting the corresponding basic principle of inference to situations where minimization of those indexes between a model and some extension of the empirical measure of the data appears as its natural extension. This leads to the so called generalized boot-strap setting for which minimum divergence inference seems to replace Maximum Likelihood one.