Influence diagram

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In decision analysis, an influence diagram (ID) (also called a relevance diagram or decision network) is a graphical and mathematical representation of probabilistic inference and decision problems. IDs are a generalization of decision trees.

An ID is a directed acyclic graph with three types of node — decision node, utility node and chance node — and two types of arrows (or arcs) between nodes — conditioning arrow (ending in a chance node) and informational arrow (ending in a decision node). A conditioning arrow indicates that the uncertain variable represented by the chance node at its head is probabilistically dependent on the outcome of the node at its tail. An informational arrow indicates that the decision at its head is made with perfect knowledge of the outcome of the variable represented by the node at its tail. Often, a single chance node in an ID is identified as its value node.

In an ID, sets such as predecessors, successors, and direct (immediate) predecessors and successors of a node are defined in the obvious manner. 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). When a WFID contains at least one decision node directly or indirectly influencing a value node, it is referred to as a well-formed decision influence diagram (WFDID), often shortened to decision diagram. WFIDs can be evaluated using reversal and removal operations to yield answers to a large class of probabilistic, inferential, and decision questions. WFDIDs can also be evaluated to produce a recommended course of action.

The term relevance diagram is perhaps a better use of language than influence diagram. An arrow 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 arrow having a defined direction. Since arrows 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 relevance diagram has never taken off, and the world continues to refer to influence diagrams. 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.

[edit] 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.