Comparing Averaged-Out Utilities of Probability Trees Having Random Parameters

This paper develops a procedure to approximate the marginal distributions of each of the following performance measures defined on a forest of probability trees with random parameters: (a) the averaged-out utility of each individual tree in the forest, and (b) the difference between the averaged-out utilities of each pair of trees in the forest. The parameters of a probability tree are its branching probabilities and node utilities, and they may be specified as mutually independent random variables with given distributions; moreover, any of these parameters may be duplicated at several places in the forest to represent parameter dependencies within and between trees. The approximation procedure incorporates an efficient,numerically robust algorithm for computing up to the first four moments of the selected performance measure. The desired distribution is approximated by an Edgeworth expansion based on the computed moments. With respect to accuracy and computational complexity, the approximation procedure is compared to estimation techniques based on simulation and normal distribution theory. To illustrate how the analysis of probability trees can directly incorporate uncertainty or random variation in the values of tree parameters, the approximation procedure is applied to the comparison of alternative medical protocols for managing patients following myocardial infarction (heart attack). In addition to its applications in medical decision making, the procedure developed in this paper should also have applications in risk analysis, reliability, project planning, marketing, and other fields in which probability trees are used to model various processes of interest.