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.
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 
-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 .
The above example highlights the power of the influence diagram in representing an extremely important concept in decision analysis known as the value of information. Consider the following three scenarios;
- Scenario 1: The decision-maker could make their 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 their 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 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 the value of information on Weather Forecast, which is essentially the 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 they should be willing to pay for such move is called the 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 their patients, diseases, etc.
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