Influence diagram
In decision analysis, an influence diagram (ID) (also called a decision network) is a compact graphical and mathematical representation of a decision situation. It is called relevance diagram (or Bayesian network) when used in probabilistic inference in which decisions are not represented in the diagram. IDs can be also be viewed as a generalization of decision trees.
An ID is a directed acyclic graph with three types of node and three types of arc (or arrrow) between nodes. As for nodes, a decision node (corresponding to each decision to be made) is drawn as a rectangle, an uncertainty node (corresponding to each uncertainty to be modeled) is drawn as an oval, and a value node (corresponding to each component of additively separable utility function) is drawn as an octagon (or diamond). As for arcs, functional arcs (ending in value node) indicate that the value node at their heads represents a function of some or all the nodes at their tails. Conditional arcs (ending in uncertainty node) indicate that the uncertainty at their heads is probabilistically dependent on (or probabilistically relevant to) some or all the nodes at their tails. Informational arcs (ending in decision node) indicate that the decision at their heads is made with perfect knowledge of the outcome of all the nodes at their tails.
With the nodes and arcs in an ID, sets such as predecessors, successors, and direct (immediate) predecessors and successors of a node are defined in the obvious manner. In addition to the acyclicity of the influence diagram, it is generally assumed that value nodes have no successor nodes. If the nodes in the influence diagram are connected with the arcs following the above conditions, it follows semantically that every node is independent on (irrelevant to) its non-successor nodes given the outcome of its immediate predecessor nodes are known.
Influence diagrams are hierarchical and can be defined either in terms of their structure or in greater detail in terms of the functional and numerical relation between diagram elements. An ID that is consistently defined at all levels—structure, function, and number—is a well-defined mathematical representation and is referred to as a well-formed influence diagram (WFID). WFIDs can be evaluated using reversal and removal operations to yield answers to a large class of probabilistic, inferential, and decision questions. More recent techniques have been developed by artificial intelligence community with their works around Bayesian network inference.
The term decision diagram is perhaps a better use of language than influence diagram. An arc connecting node A to B implies not only that "A is relevant to B", but also that "B is relevant to A" (i.e., relevance is a symmetric relationship). The word influence implies more of a one-way relationship, which is reinforced by the arc having a defined direction. Since arcs are easily reversed, this "one-way" thinking that somehow "A influences B" is incorrect (the causality could be the other way round). However, the term decision diagram is never adopted in larger community, and the world continues to refer to influence diagram. We are stuck, therefore, with a less-than-perfect nomenclature.
A complement to influence diagrams is morphological modelling which is based on a multi-dimensional configuration space linked by way of logical relationships rather than causal or probabilistic relationships.
Bibliography
- Holtzman, Samuel, Intelligent Decision Systems (1989), Addison-Wesley.
- Howard, R.A., and J.E. Matheson, "Influence diagrams" (1981), in Readings on the Principles and Applications of Decision Analysis, eds. R.A. Howard and J.E. Matheson, Vol. II (1984), Menlo Park CA: Strategic Decisions Group.
- Shachter, R.D. (1986). Evaluating influence diagrams. Operations Research, 34:871--882.
- Shachter, R.D. (1988). Probabilistic inference and influence diagrams. Operations Research 36: 589-604.
- Detwarasiti, A. and R.D. Shachter (2005). Influence diagrams for team decision analysis. Decision Analysis 2(4): 207-228.