The Quantification of Uncertainty
2.9 DECISION TREES AND INFLUENCE DIAGRAMS FOR RISK ANALYSIS
2.9.2 The Influence Diagram
Whereas decision trees depict the many scenarios of a decision problem (including the time sequence of events and decisions), they suffer from the disadvantage that they become very large, even for modestly sized situations. The size of a decision tree, as measured by the number of its terminal branches, increases exponentially with the number of variables in the problem.
DECISION OUTCOME
1
1
Consequences of 1 & O1
Consequences of 1 & O2 Consequences
of 1 & O3 O1
O2 O3
OUTCOME 2
O1
O4
Consequences of 2 & O1
Consequences of 2 & O3 Consequences
of 2 & O4 DECISION TREE
2
Utility/Cost Estimating Tools
Failure Data Analysis Tools Expected
Utility
Expected Utility
O3
Utility/Cost Estimating Tools
Probabilities
Fault Tree Analysis
Reliability Operational
Data
Expert Judgment
Maintenance Records
Figure 2.1 A decision tree for safety risk management (abstracted from NRR (1998)).
Another limitation of decision trees is that they do not explicitly depict the interdependencies between the various nodes that appear in a decision problem. Influence diagrams overcome these limitations of decision trees. They depict the decision problem as a compact graph, whose size, as measured by the number of its nodes, grows linearly with the number of variables. All the same, decision trees and influence diagrams are isomorphic; that is, any properly constructed influence diagram can be converted to a decision tree, and vice versa. A common strategy is to start off by using an influence diagram to help understand the major elements of a problem and its interdependencies, and then to convert it to a decision tree for its systematic solution.
It has been the experience of many practitioners that communicating the elements of a decision problem via influence diagrams is more effective than doing so via decision trees. However, it is also the experience of many that drawing an influence diagram is much more involved than drawing a decision tree. An expository tutorial on influence diagrams, together with several examples of their applications, is in Clemen (1991); a more mathematical treatment is in Barlow (1988) and in Barlow and Pereira (1990). The material which follows has been abstracted from these sources.
As is done with decision trees, the elements of a decision problem (namely the decisions to be made), the uncertain outcomes and the consequences of a decision–outcome combination show up in the influence diagram as different shapes. These shapes, called nodes, are then linked up by arrows to show the interrelationships between the elements. A decision node is indicated by a
DECISION TREES AND INFLUENCE DIAGRAMS FOR RISK ANALYSIS 37 rectangle and a random node by a circle; the consequence node, also called the value or payoff, is denoted by two concentric rectangles with rounded corners (Figure 2.2). The arrows connecting the nodes are called arcs, and the entire display is known as a graph. The node at the beginning of an arc is called a predecessor, and the node at the end of an arc is called a successor. A node with no adjacent predecessors is called a root node, and a node with no adjacent successors is called a sink node. A path between two nodes, say1and2, is a collection of arcs that leads one from 1 to2 via intermediate connecting nodes. In Figure 2.2, the node labeled is a root node, whereas the node labeled ‘Value’ is a sink node. Also, node R is a successor to the node labeled, which is a predecessor to.
The simplest decision problem is one in which there is only one decision to make, one uncertain event and one outcome that is determined by both the decision and the uncertain event. Thus, for example, consider a simplified version of the cholesterol drug problem considered before, in which the only decision to be made at nodeis or () and the only uncertain event of interest isorat nodeR; that is, we ignore the issue of the drug’s side effects. An influence diagram for this decision problem is shown in Figure 2.2. Observe that, since both the decision node and the random node precede the value node and also influence it, there are arcs going from these nodes to the value node. Also, since the decision to administer the drug would influence the occurrence or not of a heart attack, there is an arc going from the decision node to the random node. The absence of an arc from the random node to the decision node reflects the fact that, when the decision is made, we do not know if the patient will suffer a heart attack or not. Any arc going from a random node to a decision node indicates the fact that, when the decision is made, the outcome of the (predecessor) random node is known; such arcs are usually denoted by dotted lines. Also denoted by dotted lines are arcs going from one decision node to another; these indicate the fact that the first decision is made before the second. Finally, a well-constructed influence diagram should have no cycles of all solid lines; that is, once we leave a node we cannot get back to it, and the diagram must have at least one root node and one sink node.
or ()
or () Value
: Administer Drug (): Do Not Administer Drug : Patient Suffers Heart Attack (): Patient Avoids Heart Attack Key:
Figure 2.2 Influence diagram for a simplified version of the cholesterol drug problem.
1
Condition
2
Positive or Negative
or ()
or () Value
Figure 2.3 Influence diagram for the cholesterol drug problem with imperfect information.
The influence diagram of Figure 2.2 can be expanded to include the commonly occurring situation of imperfect information which decision makers often have prior to making a decision.
With respect to the cholesterol drug problem of Chapter 1, suppose that we have the benefit of an inexpensive medical test on the general health of the patient which gives us added but inconclusive information about the patient’s susceptibility to a heart attack. In Figure 2.3, the
() *
*
()
() Result
of Test
1
2
3
4
Figure 2.4 Decision tree for the cholesterol drug problem with imperfect information.
DECISION TREES AND INFLUENCE DIAGRAMS FOR RISK ANALYSIS 39 general health of the patient is indicated by a random node 1, and the additional test by a random node 2. This latter node is a random node because the test may reveal whether the patient is risking a heart attack (test positive) or not (test negative). The results of the test would of course depend on the true condition of the patient, and thus we have an arc going from1
to2. Furthermore, when the decision to administer the drug is made, the result of the test is known. Thus, we have a dotted arc going from 2 to (Figure 2.3). Note that the situation described above is different from that in which we have the option of deciding whether to order the test or not. In the latter case, we would have an additional decision node pertaining to the test and preceding node.
To see how an influence diagram displays features of interdependence between the nodes that the decision tree does not, we show in Figure 2.4 a decision tree that is isomorphic to the influence diagram of Figure 2.3. Observe that, unlike the arc going fromtoin the influence diagram, there is nothing in the decision tree which graphically portrays the fact that our decision influences the outcome. However, nodeof the influence diagram corresponds to nodes∗ and∗ of the decision tree to indicate the fact that our decision could influence the outcome.
Similarly, the decision tree has no analog for the arc from1to Rof the influence diagram, which reflects the fact that the outcome of the test depends on the health of the patient.
Lauritzen and Spiegelhalter (1988) discuss the use of influence diagrams in medical expert systems, whereas Good (1961) has used a similar vehicle to illustrate notions of causality.
The examples given here do not illustrate the fact that influence diagrams are generally more compact than decision trees. This feature is better appreciated in the context of sequential decision problems which have a tendency to grow exponentially. The fact that influence diagrams are useful aids for decision making vis-à-vis communication dependencies needs no further elaboration. However, the fact that they are generally difficult to construct and that they do not supplant decision trees in their entirety makes them less of a panacea than what their proponents have us believe.
Probabilistic influence diagrams
A probabilistic influence diagramis a special influence diagram in which all the nodes are random and, as before, the arcs between the nodes indicate their possible dependencies. If there is no arc connecting two nodes, then these nodes are judged to be conditionally independent, given the states of their adjacent predecessor nodes. Also, any two root nodes in a probabilistic influence diagram are independent. Associated with each node is a conditional probability for the node, and this probability depends on the states of the adjacent predecessor nodes, if any.
Probabilities associated with the root nodes are conditioned on the background information. Given a probabilistic influence diagram, there exists a unique joint probability function corresponding to the random quantities represented by the nodes. This joint probability is the product of the probabilities associated with all the nodes in the diagram. Consequently, in addressing practical problems, it may be easier to use an influence diagram to assess the joint probability distribution by multiplying the node probabilities, as opposed to a direct probability assessment.
Probabilistic influence diagrams are in essence a pictorial depiction of the calculus of proba- bility, and as such have been used by some to ensure that the laws of probability are observed.
That is, probabilistic influence diagrams are isomorphic with the calculus of probability and serve as an aid for ensuring coherence. This isomorphism is achieved by introducing the three operations, ‘node splitting’, ‘node merging’ and ‘arc reversal’, all of which are derived from the addition and multiplication rules and Bayes’ Rule. The following example in forensic science, taken from Barlow and Pereira (1990), is illustrative. Scenarios involving medical diagnosis and machine maintenance can be seen as alternate versions of this example.
An archetypal problem in forensic science goes as follows. A robbery has been committed by breaking a window and, in the process, a blood stain has been left by the robber. An individual
with the same blood type as that on the window stain has been charged with the crime. Based on this evidence, we need to assess the probability of the individual’s guilt. Let12denote the blood type of the individual (window stain), and letdenote the ‘culpability’ of the individual, with=10implying guilt (innocence). Leti=10 i=12, if the blood type is(not).
We need to evaluateP=11=2=1. The probabilistic influence diagram of Figure 2.5 describes the probability model for this problem.
Observe that the diagram does not portray the actual values of theis that are known to us. It merely describes the dependence relationships among the quantities and the probabilities to be used. Specifically, ifprepresents the proportion (chance) of persons in the population having blood typeA, andqour prior probability that the suspect is guilty of the crime, whereqhas been assessed before we learn of the blood type evidence, thenP1=1p=p, andP=1q=q.
The probabilitiespandqare assessed based on background history alone, and thusand1
go to define the root nodes1and2; note thatp=1−q. A knowledge ofphelps us assess the conditional probability of2. Specifically, we can see that
P21=p if=2=1
=1−p if=2=0
=1 if=1 and1=2 and
=0 otherwise.
Thus all the probabilities associated with the nodes of Figure 2.5 can be assessed.
Because1and2are root nodes, the events and1are judged independent, and thus P12=P21 P1 P
by the multiplication rule. Therefore, we see that the joint probability of all the events in a probabilistic influence diagram is simply the product of probabilities associated with the nodes.
1
Event
3
Event
2
Event () = q
(2 | 1, )
(1) = p
Figure 2.5 Probabilistic influence diagram for a problem in forensic science.
DECISION TREES AND INFLUENCE DIAGRAMS FOR RISK ANALYSIS 41
To obtainP=11=2=1we use the multiplication rule where P12=P12
P12
and proceed to evaluateP12. The evaluation of this latter quantity gives us an opportunity to illustrate thenode splittingoperation of influence diagrams. Since
P12=P1 P21=P2 P12
there are two ways in which the node representing the event1 2can be split (Figure 2.6).
The choice of how to split a node depends on our ability to assess the ensuing probabilities.
For example, if we choose to go with the third box of Figure 2.6, then we are required to assess P12, the probability that an individual having blood typeAis liable to commit a robbery and, in so doing, get hurt. That is, we need the probability that persons of blood type Aare sloppy thieves. Similarly, the node representing the event 12can be split in six different ways, two of which are shown in Figure 2.7.
The second probabilistic influence diagram operation isnode merging. This operation is the reverse of node splitting, and a simple way to appreciate this operation is to look at the first two boxes of Figure 2.7 in reverse order. That is, we can go from the two nodes of the second box to the single node of the first box. The same is also true of the third box. However, it is not always possible to merge any two adjacent nodes in a probabilistic influence diagram. In general, two nodes, say1and2, can be merged into a single node1 2only if there is a list ordering of the nodes such that1is an immediate predecessor or successor of2in the list. For example, in the diagram of Figure 2.8, the two nodes and2 cannot be merged because the only list
Event Event
1 2
1 2
Event
Event Event
(1 | 2) (2 | 1)
(1, 2)
(1) (2) Figure 2.6 An illustration of node splitting.
Event
(1, 2)
Event Event
(1, 2)
Event Event
(1, 2) (|1, 2) (1, 2 |) (1, 2,3)
() Figure 2.7 Another illustration of node splitting.
(1, 2) (1, 2)
Event
Event Event Event Event
2
1
Event
Event
(|1, 2) (1, 2 |)
(1, 2) ()
Figure 2.8 Illustration of node merging.
ordering here is<1<2, andand2are not neighbors on the list. We are, of course, able to mergeand1into 1and1and2into (1 2).
The third probabilistic influence diagram operation isarc reversal. This operation corresponds to Bayes’ formula, which pertains to inverting probabilities. To see how this works, consider the first box of Figure 2.9, which contains two nodes1and2with an arc from1to2. Using the node merging operation, we merge the nodes corresponding to the events1and2 to obtain the single node box in the center of Figure 2.9. We then apply the node splitting operation to
(1, 2) Event
Event Event
2
1
(2 |1) (1)
Event
2
1
Event
(1 |2) (2) Figure 2.9 Illustration of arc reversal operation (Bayes’ law).
DECISION TREES AND INFLUENCE DIAGRAMS FOR RISK ANALYSIS 43 the single node of the center box to obtain the third box of Figure 2.9, which has two nodes1
and2 with an arc going from2 to1. The arcs in the first and the third boxes are reversed.
In order to obtain the probabilitiesP12andP2we use Bayes’ law and the law of total probability, respectively. We may thus interpret the law of total probability and Bayes’ law as algebraic operations that enable us to go from the first box to the third without going through the middle box, the first and third boxes entailing a reversal of arcs.
Theorems pertaining to the conditions under which node merging and arc reversal can be undertaken are given by Barlow and Pereira (1990), who also describe the visual force of the probabilistic influence diagrams to explain the notions of conditional independence. We find them interesting, but leave it up to the reader to decide on their usefulness. Our inclusion of probabilistic influence diagrams is for the sake of completeness.