Constructing Influence Diagrams

Influence diagrams provide a visual perspective of the decisions, uncertain events (states of nature), and consequences and the subsequent interrelationships that exist within a decision problem. An appropriate model ensures the probabilistic structure of a decision problem, the timing of available information and interdependence of decisions that can be taken and which may arise under certain states of nature, in a compact form (Marshall and Oliver, 1995).

Model building of decision problems as seen can become quite complex; however, influence diagrams hold many advantages. First, they provide a framework by which a decision maker can reject or confirm assumptions and accurately model dependencies through a graphical display. Secondly, complex decision problems have the innate ability to become “messy”. Influence diagrams provide a format whereby the large volume of information can be summarized into relevant and sufficient matters. From a practical viewpoint, they provide alternative situations and are easily interpretable. The use of algorithms and numerical techniques also enable an efficient and simple analysis.

The design and construction of influence diagrams are based on the aforementioned three basic elements – decision nodes, chance (states of nature) nodes, and consequence nodes, connected by arcs. A node at the beginning of an arc is qualified as a predecessor; and consequently, a node at the end of an arc is referred to as a successor. The rules for using arcs to represent relationships among the nodes are shown in figures 9-3 and 9-4. In general, an arc can represent either relevance or sequence (Clemen, 1996). The direction of the arc indicates the meaning and brings into the relevance or sequence context.

Figure 9-3:

For example, the figure above illustrates an arrow pointing into a chance node, indicating relevance. This demonstrates that the predecessor is relevant for assessing the chances associated with the uncertain event. In the diagram above, the first cluster of nodes shows an arrow (arc) from A to C. This suggests that the chances (states of nature) associated with C may be different for different outcomes of A. Likewise, an arrow pointing from a decision node to a chance node means that the chosen decision is relevant for assessing the chances associated with the succeeding uncertain event. For instance,

Relevance

A

C

E B

D

F

the choice taken in decision B is relevant for assessing the chances associated with C’s possible outcomes. Relevance arcs can also point into consequence or calculation nodes, indicating that the consequence or calculation depends on the specific outcome of the predecessor node. The second cluster of nodes shows that consequence node F depends both on decision D and event E.

Figure 9-4:

When the decision maker has a choice to make, the choice would normally be made on the basis of information available at the time. Arrows that point to decisions represent information available at the time of the decision and hence represent sequence (figure 9-4). Such an arrow indicates that the decision is made knowing the outcome of the predecessor node. An arrow from a chance node to a decision means that from a decision maker’s point of view, all uncertainty associated with a chance event is resolved and the outcome is known when the decision is made. Thus, information is available to the decision maker regarding the event’s outcome. This is the case, as illustrated in the figure above, with chance node H and decision node I. The decision maker waits to learn the outcome of H before making decision I. An arrow from one decision to another decision simply means that the first decision is made before the second, such as decision nodes G and I. Thus, the sequential ordering of decisions is shown in an influence diagram by the path of arcs through the decision nodes.

A simple procedure for constructing an influence diagram to model the structure of a decision problem is stated in Decision Making and Forecasting by Marshall and Oliver (1995). These sequential steps are summarized below.

1. Create a preliminary list of decisions and random events (or quantities of interest) whose outcomes are believed to be important in the formulation of the problem.

Identify the attributes and objectives that are to be used to measure the consequence of the decisions and outcomes.

2. Name each random quantity and decision. Represent each random quantity with a circular node and a decision with a square node. Draw them in order of occurrence from left to right.

3. Identify any influences or dependencies between random quantities and decision.

Insert directed arcs between nodes that influence one another with the direction corresponding to the natural influence believed.

H

I G

Sequence

4. Determine any conditional independencies and represent them accurately.

5. Check to see that there are no directed cycles – i.e. a connected set of arcs in a directed path that leads out of one node and back itself.

6. Check that there any decision node that occurs before a later decision node has a directed arc from the former to the latter, Similarly, a chance node known to a given decision mode must be known to a later decision node. This is a requirement of the principle of coherence (Marshall and Oliver, 1995).

This process is fairly simple and thus alternative diagrams can be easily drawn. Clemen (1996) offers some other remarks on the construction of influence diagrams. First, the nature of the arc-relevance or sequence can be ascertained by the context of the arc within the diagram. Secondly, as stated in 5 of the procedure above, an influence diagram should not contain any cycles. Furthermore, although the construction of an influence diagram may be technically correct, there is no clear cut supposition to suggest that the influence diagram constructed is the only correct one. Thus, it cannot be stated that there is a unique correct diagram but that there are many ways in which a diagram can appropriately represent a decision problem. The representation that is the most appropriate is the one that is requisite for the decision maker. That is, a requisite model contains everything that the decision maker considers important in making the decision (Raiffa and Schlaifer, 2000). Incorporating all of the important concerns¹ of the decision is the only way to get a requisite model and adequate representation of the problem.