Existence of hidden variables having only upper probabilities

We prove the existence of hidden variables, or, what we call generalized common causes, for finite sequences of pairwise correlated random variables that do not have a joint probability distribution. The hidden variables constructed have upper probability distributions that are nonmonotonic. The theorem applies directly to quantum mechanical correlations that do not satisfy the Bell inequalities.It is a pleasure to dedicate this paper to Karl Popper in celebration of this 90th birthday. The first author has known Popper for more than three decades, and has profited much from their discussion of many different topics, among which have been the foundations of probability and the foundations of quantum mechanics, both central to the present paper.