Uncertainty Quantification and Cumulative Distribution Function: How are they Related?

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 a 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 CDF to see which 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 deal with the problem. We focus the discussion on three main cases: (1) to use mean and standard deviation to construct an envelope with them; (2) to use coefficient of variation; (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 (to replace the CDF for a small set of statistics) may indeed work in some cases. But 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 what set one chooses. So 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. So, let us safely continue to use the CDF while a good reduction is not found!