Information combination and transformation: a paradox and its resolution

Abstract

In his recent paper (https://doi.org/10.1016/j.measen.2022.100416), Willink showed two contradictory theoretical solutions to the problem of combining and transforming two sets of information about the same quantity. Each set of information is represented by a probability density function (PDF), and the transformation function is nonlinear. We refer to Willink’s contradictory result as “Willink paradox”. Two operations: (a) information combination and (b) information transformation are both mathematically valid according to probability theory. Therefore, the two contradictory solutions are both theoretically correct. In practice such as measurement uncertainty analysis, however, we cannot use both solutions; we must choose one or the other. We propose an entropy metric that can be used to provide a practical solution to the Willink paradox. Three examples are presented to illustrate the Willink paradox and the proposed entropy metric.