Aninfluence diagram (ID) (also called arelevance diagram,decision diagram or adecision network) is a compact graphical and mathematical representation of a decision situation. It is a generalization of aBayesian network, in which not onlyprobabilistic inference problems but alsodecision making problems (following themaximum expected utility criterion) can be modeled and solved.
ID was first developed in the mid-1970s bydecision analysts with an intuitive semantic that is easy to understand. It is now adopted widely and becoming an alternative to thedecision tree which typically suffers fromexponential growth in number of branches with each variable modeled. ID is directly applicable inteam decision analysis, since it allows incomplete sharing of information among team members to be modeled and solved explicitly. Extensions of ID also find their use ingame theory as an alternative representation of thegame tree.
An ID is adirected acyclic graph with three types (plus one subtype) ofnode and three types ofarc (or arrow) between nodes.
Nodes:
Decision node (corresponding to each decision to be made) is drawn as a rectangle.
Uncertainty node (corresponding to each uncertainty to be modeled) is drawn as an oval.
Deterministic node (corresponding to special kind of uncertainty that its outcome is deterministically known whenever the outcome of some other uncertainties are also known) is drawn as a double oval.
Value node (corresponding to each component of additively separableVon Neumann-Morgenstern utility function) is drawn as an octagon (or diamond).
Arcs:
Functional arcs (ending in value node) indicate that one of the components of additively separable utility function is a function of all the nodes at their tails.
Conditional arcs (ending in uncertainty node) indicate that the uncertainty at their heads isprobabilistically conditioned on all the nodes at their tails.
Conditional arcs (ending in deterministic node) indicate that the uncertainty at their heads is deterministically conditioned on all the nodes at their tails.
Informational arcs (ending in decision node) indicate that the decision at their heads is made with the outcome of all the nodes at their tails known beforehand.
Given a properly structured ID:
Decision nodes and incoming information arcs collectively state thealternatives (what can be done when the outcome of certain decisions and/or uncertainties are known beforehand)
Uncertainty/deterministic nodes and incoming conditional arcs collectively model theinformation (what are known and their probabilistic/deterministic relationships)
Value nodes and incoming functional arcs collectively quantify thepreference (how things are preferred over one another).
Alternative, information, and preference are termeddecision basis in decision analysis, they represent three required components of any valid decision situation.
Formally, the semantic of influence diagram is based on sequential construction of nodes and arcs, which implies a specification of all conditional independencies in the diagram. The specification is defined by the-separation criterion of Bayesian network. According to this semantic, every node is probabilistically independent on its non-successor nodes given the outcome of its immediate predecessor nodes. Likewise, a missing arc between non-value node and non-value node implies that there exists a set of non-value nodes, e.g., the parents of, that renders independent of given the outcome of the nodes in.
Simple influence diagram for making decision about vacation activity
Consider the simple influence diagram representing a situation where a decision-maker is planning their vacation.
There is 1 decision node (Vacation Activity), 2 uncertainty nodes (Weather Condition, Weather Forecast), and 1 value node (Satisfaction).
There are 2 functional arcs (ending inSatisfaction), 1 conditional arc (ending inWeather Forecast), and 1 informational arc (ending inVacation Activity).
Functional arcs ending inSatisfaction indicate thatSatisfaction is a utility function ofWeather Condition andVacation Activity. In other words, their satisfaction can be quantified if they know what the weather is like and what their choice of activity is. (Note that they do not valueWeather Forecast directly)
Conditional arc ending inWeather Forecast indicates their belief thatWeather Forecast andWeather Condition can be dependent.
Informational arc ending inVacation Activity indicates that they will only knowWeather Forecast, notWeather Condition, when making their choice. In other words, actual weather will be known after they make their choice, and only forecast is what they can count on at this stage.
It also follows semantically, for example, thatVacation Activity is independent on (irrelevant to)Weather Condition givenWeather Forecast is known.
The above example highlights the power of the influence diagram in representing an extremely important concept in decision analysis known as thevalue of information. Consider the following three scenarios;
Scenario 1: The decision-maker could make theirVacation Activity decision while knowing whatWeather Condition will be like. This corresponds to adding extra informational arc fromWeather Condition toVacation Activity in the above influence diagram.
Scenario 2: The original influence diagram as shown above.
Scenario 3: The decision-maker makes their decision without even knowing theWeather Forecast. This corresponds to removing informational arc fromWeather Forecast toVacation Activity in the above influence diagram.
Scenario 1 is the best possible scenario for this decision situation since there is no longer any uncertainty on what they care about (Weather Condition) when making their decision. Scenario 3, however, is the worst possible scenario for this decision situation since they need to make their decision without any hint (Weather Forecast) on what they care about (Weather Condition) will turn out to be.
The decision-maker is usually better off (definitely no worse off, on average) to move from scenario 3 to scenario 2 through the acquisition of new information. The most they should be willing to pay for such move is called thevalue of information onWeather Forecast, which is essentially thevalue of imperfect information onWeather Condition.
The applicability of this simple ID and the value of information concept is tremendous, especially inmedical decision making when most decisions have to be made with imperfect information about their patients, diseases, etc.
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 awell-formed influence diagram (WFID). WFIDs can be evaluated usingreversal andremoval operations to yield answers to a large class of probabilistic, inferential, and decision questions. More recent techniques have been developed byartificial intelligence researchers concerningBayesian network inference (belief propagation).
An influence diagram having only uncertainty nodes (i.e., a Bayesian network) is also called arelevance diagram. An arc connecting nodeA toB implies not only that "A is relevant toB", but also that "B is relevant toA" (i.e.,relevance is asymmetric relationship).
Howard, R.A. and J.E. Matheson,"Influence diagrams" (1981), inReadings 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.
