Probability Inequalities with Imprecise Previsions

We investigate how various well known probability inequalities extend to lower and upper previsions. Our focus is especially on Markov’s, Bhatia-Davis, Jensen’s and Cantelli’s inequalities. In all such cases, imprecise versions of these inequalities are available even requiring the weak consistency notion of 2-coherence, which implies that they obtain for a large number of uncertainty models. However, stronger results may be achieved with coherent lower and upper previsions. In particular, it is possible to bound lower and upper variances. Various bounds for lower and upper covariances are also presented; while being generally not tight, they require very limited amounts of information to obtain.