Influence diagram


An influence diagram 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 can be modeled and solved.
ID was first developed in the mid-1970s by decision analysts with an intuitive semantic that is easy to understand. It is now adopted widely and becoming an alternative to the 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. Extensions of ID also find their use in game theory as an alternative representation of the game tree.

Semantics

An ID is a directed acyclic graph with three types of node and three types of arc between nodes.
Nodes:
Arcs:
Given a properly structured ID:
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.

Example

Consider the simple influence diagram representing a situation where a decision-maker is planning their vacation.

Applicability to value of information

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 is the best possible scenario for this decision situation since there is no longer any uncertainty on what they care about 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 on what they care about will turn out to be.
The decision-maker is usually better off 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.

Related concepts

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. WFIDs can be evaluated using reversal and removal operations to yield answers to a large class of probabilistic, inferential, and decision questions. More recent techniques have been developed by artificial intelligence researchers concerning Bayesian network inference.
An influence diagram having only uncertainty nodes is also called a relevance diagram. An arc connecting node A to B implies not only that "A is relevant to B", but also that "B is relevant to A".