Deriving PDFs for Interrelated Quantities: What to Do If There Is “More Than Enough” Information?

The GUM and its supplements one and two have come to be regarded as de facto standards for evaluating measurement uncertainty. The latter documents have stressed the benefits of deriving probability density functions (PDFs) for the output quantities instead of just evaluating the standard and expanded uncertainties associated with their best estimates. Those supplements, however, present a brute force numerical method for obtaining the output PDFs. In this paper, we rely instead on the use of the change-of-variables theorem as a way of obtaining analytical expressions for PDFs in the form of integrals that, at least in principle, can be numerically evaluated. But more importantly, the GUM supplements do not include situations in which the available information is more than enough, in the sense that by separately considering its various components one may arrive at concurrent PDFs for the quantity or quantities of interest. In some situations these PDFs will be largely similar, so that implementing some form of merging method would perhaps be reasonable. But it may also happen that significant discrepancies between those PDFs are apparent, in which case a modification of the measurement model might make all information appear to be consistent. Herein we review two procedures for merging concurrent and not too dissimilar PDFs. We also discuss some elements to be considered if extending the measurement model appears to be a better alternative.