An analysis of decomposition for subjective estimation in decision analysis

In decision analysis, it is frequently necessary to obtain reasonably precise estimates of unknown quantities, including both probabilities and utilities. Decomposition is often used for this purpose, under the assumption that it is preferable to direct elicitation. In this context, decomposition involves estimating the conditional means of the quantity of interest for a finite number of conditioning events, and weighting the means by the estimated probabilities of these events. We propose a model for how the precision of estimates obtained using decomposition depends on the choice of conditioning events. A novel feature of this model is that it captures how the choice of conditioning events influences both the conditional means and the conditional variances simultaneously. This makes it possible to characterize not only when decomposition is preferable to direct elicitation, but also when highly informative decompositions are preferred to less informative ones. Intuitive explanations of these results are provided. In addition, when the estimated probabilities of the conditioning events are reasonably precise, our results are consistent with conventional wisdom in decision analysis, and bear strong similarity to results on the construction of strata in stratified sampling