Monotonic influence diagrams: Extension to stochastic programming and application to probabilistic design

In this paper the theory of monotonic influence diagrams is extended to deal with uncertainty. Monotonic influence diagrams (MID) are a synthesis of influence diagrams and monotonicity analysis. Formally, a monotonic influence diagram is a directed graph consisting of nodes and arcs. The nodes represent design variables and the arcs reveal their relationships. Nodes in a MID can represent either deterministic or uncertain quantities. An arc in a MID is associated with the qualitative relation between the corresponding variables. A deterministic qualitative relation between two variables is given by the sign of the partial derivative of the function defining one of the variables with respect to the other variable. A probabilistic qualitative relation is defined in terms of a constraint on the joint probability distribution of the variables. Both deterministic and probabilistic quantities and relationships will be addressed in this paper.