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

Abstract

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.

The Cosine Error: A Bayesian Procedure for Treating a Non-repetitive Systematic Effect

“…Benefiting from those discussions, in [12] and [13] we showed that, in the metrological context, the effects of Borel's paradox are mainly revealed if further transformations of the quantities involved, designed as consistency checks, are carried out. For this reason, those inconsistencies were not noticed in our original analysis of the cosine error problem.…”

“…To obtain the PDF f W (ω) by multiplication of distributions we can proceed in either of two ways [12,Sect. 3].…”

Bayesian Analysis of a Simple Measurement Model Distinguishing between Types of Information

“…Since the same pair of likelihoods is used for successive updating in (20) and (21), logarithmic pooling of the priors followed by updating with the two likelihoods would lead to the same outcome as if (20) and (21) were directly logarithmically pooled. This property, which additive pooling does not have, is called 'external Bayesianity' [13] and was considered in a metrological context in [14]. However, given that non-informative priors normally have only a minor impact on the outcome, pooling them will in most cases be an unnecessary hassle.…”