Joint probabilities under expected value constraints, transportation problems, maximum entropy in the mean

Here we consider an application of the method of maximum entropy in the mean to solve an extension of the problem of finding a discrete probability distribution from the knowledge of its marginals. The extension consists of determining joint probabilities when, besides specifying the marginals, we specify the expected value of some given random variables. The proposed method can incorporate constraints as the the requirement that the joint probabilities have to fall within known ranges. To motivate, think of the marginal probabilities as demands or supplies, and of the joint probability as the fraction of the supplies to be shipped from the production sites to the demand sites, thus joint probabilities become transportation policies. Fixing the cost of a transportation policy is equivalent to requiring that the unknown probability yields a given value to some random variable, and prescribing the range for each unknown may have an economical interpretation.