Evaluation of multilevel decision trees

Abstract

The evaluation of decision trees under uncertainty is difficult because of the required nested operations of maximizing and averaging. Pure maximizing (for deterministic decision trees) or pure averaging (for probability trees) are both relatively simple because the maximum of a maximum is a maximum, and the average of an average is an average. But when the two operators are mixed, no simplification is possible, and one must evaluate the maximization and averaging operations in a nested fashion, following the structure of the tree. Nested evaluation requires large sample sizes (for data collection) or long computation times (for simulations). An alternative to full nested evaluation is to perform a random sample of evaluations and use statistical methods to perform inference about the entire tree. We show that the most natural estimate is biased and consider two alternatives: the parametric bootstrap and hierarchical Bayes inference. We explore the properties of these inferences through a simulation study.