An influence diagram (ID) (also called a decision network) is a compact graphical and mathematical representation of a decision situation. It is a generalization of Bayesian network where not only probabilistic inference problems but also decision making problems (following maximum expected utility criterion) can be modeled and solved. A bayesian network (or a belief network) is a probabilistic graphical model that represents a set of variables and their probabilistic dependencies. ... Bayesian inference is statistical inference in which evidence or observations are used to update or to newly infer the probability that a hypothesis may be true. ... Decision making is the cognitive process of selecting a course of action from among multiple alternatives. ... The expected utility hypothesis is the hypothesis in economics that the utility of an agent facing uncertainty is calculated by considering utility in each possible state and constructing a weighted average. ...

ID was first developed in late 70's within decision analysis community with a very 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. Decision analysis (DA) is the discipline comprising the philosophy, theory, methodology, and professional practice necessary to address important decisions in a formal manner. ... In operations research, specifically in decision analysis, a decision tree is a decision support tool that uses a graph or model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. ... In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ... Game theory is often described as a branch of applied mathematics and economics that studies situations where multiple players make decisions in an attempt to maximize their returns. ... In game theory, a game tree is a directed graph whose nodes are positions in a game and whose edges are moves. ...

Semantics

An ID is a directed acyclic graph with three types (plus one subtype) of node and three types of arc (or arrrow) between nodes. A simple directed acyclic graph In computer science and mathematics, a directed acyclic graph, also called a dag or DAG, is a directed graph with no directed cycles; that is, for any vertex v, there is no nonempty directed path that starts and ends on v. ... A labeled graph on 6 vertices and 7 edges. ... A labeled graph on 6 vertices and 7 edges. ...

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; Look up decision in Wiktionary, the free dictionary. ... This article or section does not adequately cite its references or sources. ... It has been suggested that Neumann-Morgenstern utility be merged into this article or section. ...

• 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; This article defines some terms which characterize probability distributions of two or more variables. ...

• 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 (and 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.

With the nodes and arcs in an ID, sets such as predecessors, successors, and direct (immediate) predecessors and successors of a node are defined in the obvious manner. In addition to the acyclicity of the influence diagram, it is generally assumed that value nodes have no successor nodes. If the nodes in the influence diagram are connected with the arcs following the above conditions, it follows semantically that every node is probabilistically independent on its non-successor nodes given the outcome of its immediate predecessor nodes are known. In graph theory, a vertex x in a directed graph is said to be a predecessor of a vertex y if there is a path from x to y. ... In graph theory, a vertex y in a directed graph is said to be a successor of a vertex x if there is a path from x to y. ... In probability theory, to say that two events are independent intuitively means that knowing whether or not one of them occurs makes it neither more probable nor less probable that the other occurs. ...

Example

Simple influence diagram for making decision about vacation activity

Consider the simple influence diagram representing a situation where a decision-maker is planning on her vacation. Image File history File links Size of this preview: 767 × 599 pixelsFull resolution (1281 × 1001 pixel, file size: 18 KB, MIME type: image/png) I, the creator of this work, hereby release it into the public domain. ... Image File history File links Size of this preview: 767 × 599 pixelsFull resolution (1281 × 1001 pixel, file size: 18 KB, MIME type: image/png) I, the creator of this work, hereby release it into the public domain. ...

• This ID has 1 decision node (Vacation Activity), 2 uncertainty nodes (Weather Condition, Weather Forecast), and 1 value node (Satisfaction).
• Functional arcs ending in Satisfaction indicate that it 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.
• Conditional arc ending in Weather Forecast indicates her belief that Weather Forecast and Weather Condition can be dependent (or relevant).
• 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.

The applicability of this simple ID can be tremendous, especially in medical decision making when most decisions have to be made with imperfect knowledge about patients, diseases, etc. Decision making is the cognitive process of selecting a course of action from among multiple alternatives. ...

Notes

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 (WFID). 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 community with their works around Bayesian network inference (Belief propagation). Garry Kasparov playing against Deep Blue, the first machine to win a chess game against a reigning world champion. ... Bayesian inference is statistical inference in which evidence or observations are used to update or to newly infer the probability that a hypothesis may be true. ... Belief propagation is an iterative algorithm for computing marginals of functions on a graphical model most commonly used in artificial intelligence and information theory. ...

Influence diagram having only uncertainty nodes (i.e., Bayesian network) is also called relevance diagram. This is perhaps a better use of language than influence 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" (i.e., relevance is a symmetric relationship). The word influence implies more of a one-way relationship, which is reinforced by the arc having a defined direction. Since arcs are easily reversed, this "one-way" thinking that somehow "A influences B" is incorrect (the causality could be the other way round). However, the term relevance diagram is never adopted in larger community, and the world continues to refer to influence diagram. We are stuck, therefore, with a less-than-perfect nomenclature. Relevance is a term used to describe how pertinent, connected, or applicable some information is to a given matter. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

A complement to influence diagrams is morphological modelling which is based on a multi-dimensional configuration space linked by way of logical relationships rather than causal or probabilistic relationships. Morphological analysis is a technique developed by Fritz Zwicky (1966, 1969) for exploring all the possible solutions to a multi-dimensional, non-quantified problem complex. ...

Bibliography

• Detwarasiti, A. and R.D. Shachter. (2005). Influence diagrams for team decision analysis. Decision Analysis 2(4): 207-228.
• Holtzman, Samuel, Intelligent Decision Systems (1989), Addison-Wesley.
• Howard, R.A. and J.E. Matheson, "Influence diagrams" (1981), in Readings 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.
• Koller, D. and B. Milch. (2003). Multi-agent influence diagrams for representing and solving games. Games and Economic Behaviors, 45: 181-221.
• Shachter, R.D. (1986). Evaluating influence diagrams. Operations Research, 34: 871-882.
• Shachter, R.D. (1988). Probabilistic inference and influence diagrams. Operations Research 36: 589-604.

See also

• Bayesian network
• Decision tree
• Node removal
• Node reversal
• Morphological analysis

A bayesian network (or a belief network) is a probabilistic graphical model that represents a set of variables and their probabilistic dependencies. ... In operations research, specifically in decision analysis, a decision tree is a decision support tool that uses a graph or model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. ... Morphological analysis is a technique developed by Fritz Zwicky (1966, 1969) for exploring all the possible solutions to a multi-dimensional, non-quantified problem complex. ...

External links

• General Morphological Analysis: A General Method for Non-Quantified Modelling From the Swedish Morphological Society
• What are influence diagrams?