A single angler satisfaction value was estimated using these values as weights in an objective function.  Optimal decisions are defines as those that maximize or minimize the objective value. For example, the optimal decision for a waterfowl harvest decision might be to maximize harvest over some time period. In contrast, an optimal decision for a recovery plan may be to minimize extinction risk over some time period. The optimal decision then is defines as that with the greatest (or smallest depending on the objective) objective value.

Identifying decision alternatives

The next step is to identify or formulate possible alternatives for the decisions. In some instances, the alternatives will be limited by the decision situation (e.g., simple yes/no decisions), whereas others may be varied and complicated (e.g., decisions on where and to improve habitat). Often, novel alternatives can be developed if the decision-makers are willing to allow for some creativity.  For example, in above the largemouth bass example, one potential way to satisfying anglers might be to build fish-attracting structures. This built-in flexibility is one of the advantages of using decision analysis because it can lead to novel solutions and important advances in resource management. Most management decision-making, however, are probably restricted by agency mandates and regulations.

Structuring the decision

The next step, modeling, is perhaps the most difficult. During this process, the problem is broken down into smaller more manageable components and relationships among the components are determined. The decision model should be as simple as possible (i.e., have the fewest components) to facilitate analyses and interpretation, but should retain all of the components that will significantly affect the outcome of the decision. For most resource management applications, decision models will likely consist of one or more population models are perhaps, some model of habitat dynamics such, forest succession, streamflow regime, etc. At this point, it is crucial to identify all those uncertainties that are likely to affect a decision. For example,

Uncertainties

Environmental factors (Weather, water/land use)

Harvest mortality (legal and illegal)

Survival rates
Growth rates

Interspecific interaction

Killer meteor

Change in administration

Similar to the objectives, we include only those events that are relevant to the decision context (e.g., bold). One often overlooked uncertainty not included in the example list above is system uncertainty. That is, the uncertainty about how the system works. This is often a major influence determining which is the optimal decision.  For example, it is unlikely that anyone would argue that water is important to stream fish. However, a determining how much water can be used without doing harm to populations requires assumptions of the relationship between population parameters and water use. Below is shown 3 hypothetical relationships between population size and water use. Under the blue model, a significant amount of water could be withdrawn before doing harm, whereas the red model suggest that most harm is done when the first few gallons are removed.

The relationships among decision components can be graphically represented in influence diagrams or alternatively, decision trees. Influence diagrams provide explicit representations of the individual components of the decision and their probabilistic dependencies. For example, the hypothetical land acquisition decision is shown in below.  Geometrical shapes referred to as nodes represent individual components. Decision nodes are represented by rectangles; chance or uncertainty nodes, by ovals; and consequence nodes, by rectangles with rounded corners. A directed arc is used to indicate dependencies between model components. For example, both colonization potential and habitat quality influence the persistence of both species A and B. Although they resemble flowcharts, influence diagrams are fundamentally different. An influence diagram represents an instantaneous moment in time. Therefore, the arcs usually (see value of information, below) do not represent the timing or sequence of events.

Parameterizing a decision model

The next step during modeling is parameterizing the dependencies among the model components. These can be estimated using empirical models, meta-analyses, and expert judgment. We have already discussed population modeling in detail earlier. Here, we will demonstrate model building using expert judgment and probabilistic networks. Probabilistic networks, also known as Bayes networks, are influence diagrams without decision nodes and consequence (utility) nodes.  They model relationships among components using probabilistic (conditional) dependencies.  For example, consider a model of fishing where the the chances of catching a fish depending on bait staying on the hook and a fish being hungry.

In the Bayes network jargon, these components are referred to as nodes, with bait on hook and fish hungry as parent nodes and fish caught as a child node.

Let's assume that each model component consists of 2 states YES and NO, so that the bait either stays on the hook (bait on hook = YES) or it doesn't (bait on hook = NO), a fish is either hungry (fish hungry=YES) or not (fish hungry = NO), and a fish is either caught (fish caught = YES) or it isn't caught (fish caught = NO).  

The possible combinations for catching a fish or not catching a fish include:

              Bait on Hook            Fish Hungry             Catch Fish
                     YES                     YES                         YES
                     YES                     YES                          NO
                     YES                      NO                          YES
                     YES                      NO                           NO
                      NO                      YES                         YES
                      NO                      YES                          NO
                      NO                       NO                          YES
                      NO                       NO                           NO
 

Decision tree model of fishing

For convenience, these combinations are often displayed in Decision Tree.

We can now use our conceptual model shown in the decision tree to estimate the probability of catching a fish on any given cast.  First, we must estimate the probability of bait staying on the hook and the fish being hungry on any given cast.  Then we must estimate the probability of catching a fish given the various combinations of bait on hook and fish hungry.  Because we do not have data on these phenomena, we must rely on our expert judgment to develop subjective probabilities (i.e., our beliefs) as is often done in real world applications. 

The probability of catching a fish on any given cast are calculated by multiplying the the probabilities for each combination that produces a fish (catch fish = YES) and summing these. This can be done by hand or via a spreadsheet. Open fishing.xls and fill in the probabilities.

BAYESIAN BELIEF NETWORK MODEL OF FISHING

Decision trees generally become unwieldy (huge) as the number of factors and the number of states per factor increases.  A more concise means of representing the same model is with a Bayesian belief network, BBN, (aka a probabilistic network).   Open fishing.dne (a Netica file) and notice the structure of the model.

BBNs consist of a set of nodes connected by vectors (arrows) that represent dependence among nodes with child nodes dependent upon parent nodes.  Here, Bait on Hook and Fish Hungry are the parent nodes and Catch Fish is the child node.  The arrows indicate that catching a fish is dependent upon Bait on Hook and Fish Hungry.  However, Bait on Hook and Fish Hungry are independent because they have no direct relationship (no arrows between them).  Also note that these 2 nodes have no arrows pointing into them.  Thus, they are referred to as root nodes.  NOTE: BBNs NEVER CONTAIN CLOSED LOOPS!

Double click on the Bait on Hook node and click on "Table".  This is where the probabilities are entered. Root nodes have unconditional probability tables and other nodes have conditional probability tables (CPTs).  Enter the unconditional probability values for Fish Hungry and the conditional probability values for Catch Fish. Compile the model by clicking on the lightning bolt symbol at the top. 

Let's assume that we know that the weather affects fish hunger.  We can add this effect by adding an additional node to the BBN. To do so, click on the yellow oval symbol in the toolbar and click the area above the Fish Hungry.

Next, click on "Style" and choose belief bars.

Then click on the arrow symbol in the tool bar and connect the two nodes.

Double click on the new node, rename the node weather, and rename the states sunny and cloudy.

As before, add probabilities to the weather tables. You will also need to add probabilities to the CPT table for Fish Hungry.  When finished, compile the program by clicking on the lightning bolt.

The above demonstrated the concept of conditional independence. That is, once you knew that fish were hungry the weather didn't effect catching a fish. In essence, catching a fish was independent of weather when you knew the condition of fish hunger. This concept is important for several reasons but particularly for the concept of modularity.

 Adding a decision
To add a decision, click on the blue square symbol in the toolbar and click the area above the Catch Fish. Lets assume that the decision is to cast (YES, NO) and that it affects whether of not we catch a fish.

in a decision tree format it would look like

Expected value of a decision
The expected value of a decision is simply the probability-weighted average of its possible values. For example, consider a yes - no decision with two possible outcomes, A and B, with values of 10 and 100, respectively. The conditional probabilities for A and B given a yes decision (e.g., P(A | yes)) are 0.75 and 0.25 and for a no decision, 0.5 and 0.5, respectively. Thus, the expected value of a yes decision would be 0.75_*10 + 0.25_*100 = 32.5 and the expected value of a no decision, 0.5_*10 + 0.5_*100 = 55. The optimal decision is simply the one with the greatest expected value, which for this example is no. 

To add a utility, click on the green symbol in the toolbar and click the area below the Catch Fish. Let's assume that our utility is enjoyment value and that enjoyment is 100 if we catch a fish and 25 if we don't (hey, it's nice to be outside).

Here casting has the greatest enjoyment value and is the optimal decision. Estimation of objective value is best shown in tree format:

The objective value is simply the probability weighted enjoyment value.

Representing system uncertainty
System uncertainty can easily be incorporated into probabilistic networks. To illustrate, open system_uncertainty.dne and compile. Examine the model structure.  It is a land acquisition decision with 4 nodes, a decision and utility node, a node representing the predicted future status of a population and the remaining node represents the uncertainty associated with the effects of land acquisition. Clink on the system dynamics node and examine the effect on the optimal decision.

Parameterizing the probabilistic networks
There are several other means to parameterizing probabilistic networks.  One such approach is to use "expert" judgment as we have done for the fishing model.  Another approach is to model the relationships directly with data. Open LinearModel.dne and compile.  Examine the relationship between canopy density, summer precipitation and clutch size.  Data for this model is contained in ClutchData.cas. Save this file to you're disk.  Next, click on "Relation" and select "incorporate case file".  Select the clutch datafile and hit enter. Recompile the network and examine the relationships.

The networks also can be parameterized using existing models-- say for example, a linear regression model. Lets assume that we have the following regression equation from a previous study of bird clutch size.
                          clutch size = 3 + canopy density + 0.5*summer precipitation + error
and the error normally distributed with a mean 0 and variance = 1.5.  Open LinearModel.dne, double click on the clutch size node, and examine the equation.  Now, highlight all of the nodes, go to the "relation" pull down menu, select "equation to table", and recompile the network.  Examine the relationships.

 

Sensitivity analysis
The next step is to examine the existing model with sensitivity analysis. In general, sensitivity analysis is used to identify the components that are most critical to the decision and is most useful for prioritizing additional modeling and data collection efforts (e.g., monitoring).  Although there are several variations to sensitivity analysis (e.g., event and joint sensitivity analyses), the basic objective is to examine each model component and determine its relative influence on the outcome (e.g., the fish population response) or the expected value of the decision.  The most influential components are considered critical to the decision and hence, are given higher priority as potential monitoring variables.

To illustrate the basics, open the largemouth bass example, bass.dne. Clink on harvest mortality low. The optimal decision under this value is no limit with a angler satisfaction of 65.53. Now click on the largest value for harvest mortality. Here the optimal decision is a 16 in limit with a value of 33.14. Doing the same with all the components would result in the following sensitivity  analysis as displayed in a tornado diagram.

Here, the decision is most sensitive to angling mortality followed by fry carrying capacity and lest sensitive to assumptions regarding density dependent growth. This type of one-way sensitivity analysis is useful for identifying influential model components. Another important consideration, however, is the sensitivity of the optimal decision to changes in the value of each model component.  A component can have a strong influence on the value of a decision but the optimal decision may remain unchanged regardless of its value.  For the LMB model, click on the various harvest mortality levels and determine the optimal decision. Notice that it changes from "no limit" to "16 in limit" as the harvest mortality increases. Do the same with other components. In this instance, the decision choice is strongly influenced by harvest mortality.

 

Expected value of information
A first approximation of the value of collecting additional data (e.g., monitoring) can be obtained by calculating the expected value of perfect information (EVPI). EVPI is the increase in the expected value of a decision should the 'true' value of a component(s) or the relationship among components become known. Thus, it can be used, in part, to identify and rank potential variables for monitoring or additional data collection efforts. For example, open the example model, fish_examp1.dne and compile.

Here the optimal decision is low water use with an expected value of 9.65. Graphically, EVPI is represented as an arc connecting an uncertainty node(s) to a decision node(s) as shown below. These arcs represent timing or sequence and indicate that the uncertainty will be resolved (i.e., the information will be known) before the decision is made. For example, current fish population status (present/absent) influenced predicted fish population status in a water use decision model, but was assumed to be unknown.

Assuming that current fish population status could be estimated without error during a survey, the expected value of the water use decision, following a survey, is estimated as the probability-weighted expected net utility for each combination of (known) watershed slope and current population size.  In the timber harvest example, the optimal decision is none when fish population is present with expected value of 18.25 and moderate when a population is absent  with an expected value of 5.45.  Because the current fish population is unknown when the decision to conduct a survey is made, the prior probabilities of a present or absent (i.e., 0.50) must be used to calculate the expected value. For example, the expected value of 'knowing' current population status is the sum of the probability-weighted values, i.e.,

(0.50_*18.25) + (0.50_*5.45) = 11.85.

EVPI is calculated as the difference between the expected value with and without a survey,

11.85 - 9.65 = 2.20. 

Even the most carefully controlled experiments- or carefully made measurements- will have some uncertainty (e.g., variance) associated with them, which can affect their value. To account for the error in measurements or models, requires the estimation of the expected value of imperfect information (EVII). EVII is considerably more complicated to estimate than EVPI and requires an estimate of the expected outcome and the use of Bayes rule to calculate posterior probabilities and expected values.

 

 

 

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Last updated 27 May 2006