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This note studies some alternatives and weak versions of the Independence axiom in a decision theoretic framework under uncertainty. We propose a characterization of this axiom using a property called Weight Independence. Moreover we study how the Independence axiom is related with the Variational Bewley model of Faro [2015]. We show that Variational Bewley preferences satisfy a weaker form of independence called Independence for Constant Weights. This topic gives us the opportunity to discuss the pioneeristic contributions of David Schmeidler on the weakening of the Independence axiom. JEL Codes: D81.