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Health Informatics Unit
The University of Adelaide
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Bayesian Decision Models

Michael Edmonds

Bayesian decision models represent a clinical problem and the relevant probabilities, dependencies and uncertainties using influence diagrams. Influence diagrams represent a compact structure that shows the complex probabilistic relationships within a domain . The diagrams consist of various nodes, each representing a different factor (see figure). Chance nodes (oval shaped) represent uncertainty, which may be influenced by other uncertainties or by a decision that is made. The decision is shown with a decision node (rectangle). An arc shows the influence of any node on another. An arc between two chance nodes suggests that a probabilistic relationship may exist, meaning that the occurrence of one event alters the probability of the other. This is the conditional probability. A node is the parent of another node, which becomes the child node, if an arc links from the parent node to the child node. The absence of an arc between two nodes is an indicator of conditional independence, where the occurrence of one event will not alter the probability of the other.

The outcomes of the domain are measured by a utility node (diamond shaped). This utility can be any measure such as cost, quality-adjusted-life-years, patient comfort or quality of care, depending on the perspective of the user. Utility is a function of whatever outcomes contribute to these measures. Utility will vary with each decision made as the probabilities of the potential outcomes will be changed. The best decision will be that which maximises utility, such as maximum patient comfort, or minimum cost.

In the figure the prior probability of the parent node will determine the conditional probability of the child node. The decision is based on the probability of the child node, and the utility is a function of that decision, given the state of the parent node. The parent node and the decision are independent of each other. An example using this structure would of the parent node being the weather, the child node being the forecast, with the decision of whether to take an umbrella or not, against the utility of comfort. The forecast is based on the weather, but the decision to take an umbrella is based on the forecast, not the actual weather. Comfort will obviously be a function of whether you have an umbrella or not, and what the weather is.

Probabilities are determined using Bayes' theorem, based on the work of Thomas Bayes (1702-1762), an English philosopher, mathematician and theologian 2 . This theorem uses population prevalence, or prior probability, to determine the conditional probability 3-5 . Where a node does not have any arcs leading into it, the probability is the prior probability. Where the state of a parent node is unknown, its prior probability will determine the conditional probabilities of its child nodes.

Information and data to define the probabilities and outcomes of these models is ideally identified using the principles of evidence-based medicine.

References

1. Nease R, Owens D. Use of Influence Diagrams to Structure Medical Decisions. Medical Decision Making 1997;17(3):263-275.

2. Andrade PJ. Specialized computer support systems for medical diagnosis. Relationship with the Bayes' theorem and with logical diagnostic thinking. Arq Bras Cardiol 1999;73(6):545-552.

3. Bland JM, Altman DG. Bayesians and frequentists. BMJ 1998;317(7166):1151-60.

4. Gurrin LC, Kurinczuk JJ, Burton PR. Bayesian statistics in medical research: an intuitive alternative to conventional data analysis. J Eval Clin Pract 2000;6(2):193-204.

5. Oberson J. Notes on Diagnosis Reasoning. Bayesian Systems. Applications for the creation of computer-assisted tools for diagnosis. Procedures for the development of MINERVA., 1998.