Combining Probability Distributions by Multiplication in Metrology: A Viable Method?

In measurement science quite often the value of a so-called 'output quantity' is inferred from information about 'input quantities' with the help of the 'mathematical model of measurement'. The latter represents the functional relation through which outputs and inputs depend on one another. However, subsets of functionally independent quantities can always be so defined that they suffice to express the entire information available. Reporting information in terms of such a subset may in certain circumstances require aggregating probability distributions whose arguments are interrelated quantities. The option of aggregating by multiplication of distributions is shown to be susceptible of yielding inconsistent results when the roles of inputs and outputs are assigned differently to the quantities. Two alternatives to this practice that do not give rise to such discrepancies are discussed, namely (i) logarithmic pooling with weights summing to one and (ii) linear pooling, of which the former appears to be slightly more favourable for applications in metrology. An example illustrates the inconsistency of results obtained by distinct ways of multiplying distributions and the manner in which these results differ from a logarithmically pooled distribution.