The MinMax information measure

The importance of finding minimum entropy probability distributions and the value of minimum entropy for a probabilistic system is discussed. A method to calculate these when there are both moment and inequality constraints on probabilities is given and illustrated with examples. It is shown that: information given by moments or inequalities on probabilities can be measured by the reduction in the uncertainty gap (S max - S min ); and in certain circumstances the inequalities on probabilities can provide significant information about probabilistic systems.