What is uncertainty quantification?

Abstract

Uncertainty is described by the cumulative distribution function (CDF). Using the CDF one describes all the main cases: the discrete case, the case when an absolutely continuous probability density exists, and the singular case, when it does not, or combinations of the three preceding cases. The reason one does not see any mention of uncertainty quantification in classical books, as Feller’s and Chung’s, is that they found no reason to call a CDF by another name. However, one has to acknowledge that to use a CDF to describe uncertainty is clumsy. The comparison of the CDFs of two random variables to evaluate which one of them is more uncertain is not evident. One feels that there must be a simpler way. Why not to use some small set of statistics to reduce a CDF to a simpler measure, easier to grasp? This seems a great idea, and indeed, one finds it in the literature. Several books are appearing dealing with the problem. We focus the discussion on three main cases: (1) to use mean and standard deviation to construct an envelope with them to make a nice graph; (2) to use mean and coefficient of variation; and (3) to use Shannon entropy, a number, that could allow an ordering for the uncertainties of all CDF that have entropy, a most desirable thing. The reductions consisting of replacing the CDF for a small set of statistics may indeed work in some cases. However, they do not always work, and moreover, the different measures they define may not be compatible. That is, the ordering of uncertainty may vary depending upon what set one chooses. Therefore, the “great idea” does not work so far, but they are happily used in the literature. One of the objectives of this paper is to show, with examples, that the three reductions used to “measure” uncertainties are not compatible. The reason it took so long to find out the mistake is that these reductions methods are applied to very complex problem that hide well the unsuitability of the reductions. Once one tests them with simpler examples, one clearly sees their inadequacy. Therefore, let us safely continue to use the CDF, while a good reduction is not found!