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This paper describes conditioned Dempster-Shafer (CDS) theory, a probabilistic calculus for dealing with possibly non-Bayesian evidence when the underlying a priori knowledge base is possibly non-Bayesian. Specifically, we show that the Dempster-Shafer combination operator can be “conditioned” to reflect the influence of any a priori knowledge base which can be modeled by a Dempster-Shafer belief measure. We show that CDS is firmly grounded in probability theory-specifically, in the theory of random sets. We also show that it is a generalization of the Bayesian theory to the case when both evidence and a priori knowledge are ambiguous. We derive the algebraic properties of the theory when a priori knowledge is assumed fixed. Under this assumption, we also derive the form of CDS in the special case when fixed a priori knowledge is Bayesian