Influence diagram

In [decision theory]] an influence diagram (ID) (also called a relevance diagram, decision diagram or a decision network) is a compact graphical and mathematical representation of a decision situation. It is a generalization of a Bayesian network, in which not only probabilistic inference problems but also decision making problems (following maximum expected utility criterion) can be modeled and solved.

ID was first developed in mid-1970s within the decision analysis community with an intuitive semantic that is easy to understand. It is now adopted widely and becoming an alternative to decision tree which typically suffers from exponential growth in number of branches with each variable modeled. ID is directly applicable in team decision analysis, since it allows incomplete sharing of information among team members to be modeled and solved explicitly. Extension of ID also find its use in game theory as an alternative representation of game tree.

Semantics
An ID is a directed acyclic graph with three types (plus one subtype) of node and three types of arc (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 separable Von 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 is probabilistically 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 the alternatives (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 the information (what are known and their probabilistic/deterministic relationships)
 * Value nodes and incoming functional arcs collectively quantify the preference (how things are preferred over one another).

Alternative, information, and preference are termed decision 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 $$d$$-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 $$X$$ and non-value node $$Y$$ implies that there exists a set of non-value nodes $$Z$$, e.g., the parents of $$Y$$, that renders $$Y$$ independent of $$X$$ given the outcome of the nodes in $$Z$$.

Example


Consider the simple influence diagram representing a situation where a decision-maker is planning her 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 in Satisfaction), 1 conditional arc (ending in Weather Forecast), and 1 informational arc (ending in Vacation Activity).


 * Functional arcs ending in Satisfaction indicate that Satisfaction is a utility function of Weather Condition and Vacation Activity. In other words, her satisfaction can be quantified if she knows what the weather is like and what her choice of activity is.  (Note that she does not value Weather Forecast directly)
 * Conditional arc ending in Weather Forecast indicates her belief that Weather Forecast and Weather Condition can be dependent.
 * Informational arc ending in Vacation Activity indicates that she will only know Weather Forecast, not Weather Condition, when making her choice. In other words, actual weather will be known after she makes her choice, and only forecast is what she can count on at this stage.


 * It also follows semantically, for example, that Vacation Activity is independent on (irrelevant to) Weather Condition given Weather Forecast is known.

Applicability in value of information
The above example highlights the power of influence diagram in representing an extremely important concept in decision analysis known as value of information. Consider the following three scenarios;
 * Scenario 1: The decision-maker could make her Vacation Activity decision while knowing what Weather Condition will be like. This corresponds to adding extra informational arc from Weather Condition to Vacation Activity in the above influence diagram.
 * Scenario 2: The original influence diagram as shown above.
 * Scenario 3: The decision-maker makes her decision without even knowing the Weather Forecast. This corresponds to removing informational arc from Weather Forecast to Vacation 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 she cares about (Weather Condition) when making her decision. Scenario 3, however, is the worst possible scenario for this decision situation since she needs to make her decision without any hint (Weather Forecast) on what she cares about (Weather Condition) will turn out to be.

The decision-maker is usually better off (definitely no worse off) to move from scenario 3 to scenario 2 through the acquisition of new information. The most she should be willing to pay for such move is called value of information on Weather Forecast, which is essentially value of imperfect information on Weather Condition.

Likewise, it is the best for the decision-maker to move from scenario 3 to scenario 1. The most she should be willing to pay for such move is called value of perfect information on Weather Condition.

The applicability of this simple ID and the value of information concept is tremendous, especially in medical decision making when most decisions have to be made with imperfect information about patients, diseases, etc.