Editing Influence diagram (section)

==Semantics==

An ID is a [[directed acyclic graph]] with three types (plus one subtype) of [[graph node|node]] and three types of [[graph arc|arc]] (or arrow) between nodes.

Nodes:
:*''[[Decision making|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 [[conditional probability|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 <math>d</math>-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 <math>X</math> and non-value node <math>Y</math> implies that there exists a set of non-value nodes <math>Z</math>, e.g., the parents of <math>Y</math>, that renders <math>Y</math> independent of <math>X</math> given the outcome of the nodes in <math>Z</math>.