Sampling of uncertain probabilities at event tree nodes with multiple branches

When the uncertainty distributions of the conditional probabilities of a complete, mutually exclusive set of three or more branches at an event tree node are independently sampled, it is not possible that their values can both be constrained to add to 1.0, as they must, and also have their sample means converge as sampling size increases to the probabilities’ original means, as consistency with the initial point estimates that are these means demands. Complex sampling procedures may be applicable in special cases; e.g. when the uncertainty distributions are restricted to be betas that are elements of a Dirichlet distribution, and so have variances with restricted relationships to their means. Such procedures are not applicable in general and furthermore may not be practical in the analysis of large event trees. The expansion of a multiple node as a sequence of binary nodes with appropriate conditional probabilities, which is quite possible when only the point estimates are considered, has been attempted for uncertainty modeling but can be seen to be incorrect. Normalization of the sets of independent sample values of the launch probabilities is the most common procedure but obviously cannot provide the desired convergence. In this paper, a generally applicable procedure is derived for adjusting the sample values so that they correctly add to 1.0 and their means converge to the original means. The superiority of the procedure to common normalization is demonstrated in an illustrative application in a space launch vehicle risk analysis.