Combining Probabilistic Estimates to Reduce Uncertainty
Book, S.
MCR, LLC

Suppose we have contracted for or otherwise obtained n probabilistic estimates, independent of each other, of the same system or project. This means that we have, for each estimate, a probability distribution (likely something close to the lognormal), an S-curve, a mean, and a standard deviation. We want to combine these n estimates to obtain one estimate that contains less uncertainty than each of the n estimates individually. There are two questions that we have to answer: (1) How should we "combine" the estimates, and (2) Will the combined estimate actually be less uncertain than each of the n independent estimates individually. In order for this issue to be meaningful, we must assume that each of the estimates is "correct," i.e. (1) they are neither too optimistic, nor too pessimistic, but are based on risk assessments validly drawn from the same risk information available to each estimating team; (2) each estimating team has applied appropriate mathematical techniques to the cost-risk analysis, including (for example) inter-element correlations when appropriate; and (3) each estimating team was working from the same ground rules, but may have applied different estimating methods and made different assumptions when encountering the absence of some required information.

In mathematical terminology, if we have three probabilistic estimates of a system or project whose three means are μ1, μ2, and μ3, respectively, and whose coefficients of variation are λ1, λ2, and λ3, where λ1 ≤ λ2 ≤ λ3, the question we want to answer is, "Is it better to combine multiple estimates or simply to use the "best" of them? If reducing uncertainty in your estimate is our goal, the choice is between using the least uncertain of the three estimates, namely the first, or to combine the estimates and hope that the coefficient of variation λ of the combined estimate is smaller than λ1. As an example of one way to answer the question, we propose here a numerical test to determine whether the weighted (by their respective coefficients of variation) average of the three estimates has less uncertainty than the least uncertain of the original estimates.