Benavoli, Facchini and Zaffalon

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Closure operators, classifiers and desirability

At the core of Bayesian probability theory, or dually desirability theory, lies an assumption of linearity of the scale in which rewards are measured. We revisit two recent papers that extended desirability theory to the nonlinear case by letting the utility scale be represented either by a general closure operator [Miranda22] or by a binary general (nonlinear) classifier [Casanova22]. By using standard results in logic, we highlight the connection between these two approaches and show that this connection allows us to extend the separating hyper plane theorem (which is at the core of the duality between Bayesian decision theory and desirability theory) to the nonlinear case.