On the Quantification and Decomposition of Uncertainty

Abstract

In this work we deal with the quantitative assessment and decomposition of uncertainty. The decision making process is often accompanied by an uncertainty propagation exercise in the practice. We first analyze the meaning of uncertainty propagation from a subjective decision-making point of view. We show that, in order to quantify uncertainty, one has to resort to the distribution of the expected utility (U) originated from parameter uncertainty. We undertake the analytical determination of the moments U. We show that, if one considers the uncertain parameter space as subdivided in alternative preference regions delimited by indifference hypersurfaces, the moments of U are the sum of the moments of the expected utility of alternatives in the regions alternatives are preferred. As a consequence, if an alternative is never preferable, it does not contribute to uncertainty. In order to decompose uncertainty, we focus on the variance of U. By stating of Sobol’ variance decomposition theorem in the Decision-Theory framework, we show that the variance of U can be expressed as sum of the variances brought by uncertain parameters individually and/or in groups. We then determine and discuss the meaning of global importance of parameters. Since parameters associated with the highest value of the global importance are the most effective in reducing uncertainty, gathering information on these parameters would reduce uncertainty in the most effective way. We illustrate the moment calculation and variance decomposition procedures by means of an analytical example. The application to the uncertainty analysis of an industrial investment decision-making problem concludes the paper.