Intervention

The idea of accommodating external interventions was introduced above when motivating the causal interpretation of BNs. For example, the installation of an artificial aerator to a water body may completely eliminate the chances of hypoxia. This type of intervention can be represented in the network by breaking the links between algal density (A), season (S), and hypoxia (H) and keeping H fixed in a state of false (Figure 5). Alternatively, the management of the aerator can be shown explicitly by adding a node (0) that is a parent of H and can be in a state of either off or on (Figure 6). In the on state, H is false, while in the off state, H has the same conditional distribution as before the intervention. A marginal distribution is then specified for 0 to describe the management plan for the aerator. Prediction and inference on the new network including 0 can then proceed in the usual way. As

Figure 5 A BN showing the effect of an external intervention that keeps hypoxia in a state of false.

prefer the action that maximizes a particular mathematical combination of the derived probabilities and utilities.

BNs provide a convenient and appropriate tool for generating the outcome probabilities required for decision analysis. If the children of a fully specified network represent outcome variables and the root parents represent decision variables, then the necessary probabilities can be generated by forward prediction. In most management contexts, actions to be considered often involve some type of system intervention, and these can be handled appropriately as described in the previous section.

Many software packages support the inclusion of utility nodes in a BN in addition to decision nodes and chance variables. Such networks are referred to in the decision analysis literature as influence diagrams. Influence diagrams can often be solved directly to find the optimal action, either for the unconditioned network or after findings have been added for some of the network variables. This facilitates value of information analysis.

Figure 6 A BN explicitly showing the effect of adding an artificial oxygen aerator (O).

described above, the ability of a causal BN to correctly represent interventions is an essential element of its usefulness. The use of BNs for decision analysis is one reason for this, as discussed next.

Decision

Decision analysis is a normative method for selecting among actions that have uncertain outcomes. This outcome uncertainty can be characterized by probability distributions for variables that represent the key consequences of the considered actions. The decision maker's relative preference for the various possible outcomes can then be described by a utility function that also captures the decision maker's attitude toward risk. A logical decision maker should then

Special Cases Hierarchical Models

Hierarchical BNs can accommodate additional complexity in uncertainty representation by explicitly separating the parameters, hyperparameters, and data from the processes that are the usual focus of a causal network. For example, there may be multiple sources of data for any given model variable, and each of these may have its own measurement error. These sources could be added as explicit nodes in the network with corresponding parameters describing the error magnitude and/or bias (Figure 7). Further, the

Figure 7 An example of how a hierarchical BN can be used to represent complexity by allowing multiple levels of uncertainty. The processes relating the two variables nutrient concentration (N) and algal density (A) are characterized by a vector of parameters 0A. If 6a is expected to vary across space or time, this variability can be described by the hyperparameter vector 0a. If, in addition, algal density is measured using more than one method (e.g., in situ chlorophyll and remotely sensed chlorophyll), the different measurement errors can be described by parameters 6C and 0R.

Figure 7 An example of how a hierarchical BN can be used to represent complexity by allowing multiple levels of uncertainty. The processes relating the two variables nutrient concentration (N) and algal density (A) are characterized by a vector of parameters 0A. If 6a is expected to vary across space or time, this variability can be described by the hyperparameter vector 0a. If, in addition, algal density is measured using more than one method (e.g., in situ chlorophyll and remotely sensed chlorophyll), the different measurement errors can be described by parameters 6C and 0R.

Figure 8 A temporal BN, showing how algal density (A) in one time period can influence nutrient concentration (N) in the next.

parameters characterizing the causal processes in the network may be variable across space, time, individuals, or groups. This variability can be captured by conditioning these parameters on higher-level parameters, called hyperparameters. In this way, stochasticity is allowed at multiple levels, each conditioned on one level higher. Complex statistical analysis can then proceed in reduced dimensions: rather than asking, ''How does the entire process work.?'' after conditioning we can ask, ''How does this component work, conditioned on those elements directly affecting it?'' The complexity of nature then emerges when we marginalize across the components.

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Figure 9 A BN of estuarine eutrophication that integrates a number of submodels, shown as rounded squares in the main network. Parameters of the submodels are shown as shaded nodes. Reproduced from Borsuk ME, StowCA, and Reckhow KH (2004) A Bayesian network of eutrophication models for synthesis, prediction, and uncertainty analysis. Ecological Modelling 173: 224, with permission from Elsevier.

Figure 9 A BN of estuarine eutrophication that integrates a number of submodels, shown as rounded squares in the main network. Parameters of the submodels are shown as shaded nodes. Reproduced from Borsuk ME, StowCA, and Reckhow KH (2004) A Bayesian network of eutrophication models for synthesis, prediction, and uncertainty analysis. Ecological Modelling 173: 224, with permission from Elsevier.

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