The publication of the Guide to the Expression of Uncertainty in Measurement (GUM), and later of its Supplement 1, can be considered to be landmarks in the field of metrology. The second of these documents recommends a general Monte Carlo method for numerically constructing the probability distribution of a measurand given the probability distributions of its input quantities. The output probability distribution can be used to estimate the fixed value of the measurand and to calculate the limits of an interval wherein that value is expected to be found with a given probability. The approach in Supplement 1 is not restricted to linear or linearized models (as is the GUM) but it is limited to a single measurand. In this paper the theory underlying Supplement 1 is re-examined with a view to covering explicit or implicit measurement models that may include any number of output quantities. It is shown that the main elements of the theory are Bayes' theorem, the principles of probability calculus and the rules for constructing prior probability distributions. The focus is on developing an analytical expression for the joint probability distribution of all quantities involved. In practice, most times this expression will have to be integrated numerically to obtain the distribution of the output quantities, but not necessarily by using the Monte Carlo method. It is stressed that all quantities are assumed to have unique values, so their probability distributions are to be interpreted as encoding states of knowledge that are (i) logically consistent with all available information and (ii) conditional on the correctness of the measurement model and on the validity of the statistical assumptions that are used to process the measurement data. A rigorous notation emphasizes this interpretation.
The Bayesian analysis of a simple calibration model is reconsidered. Observed values are at hand that conform to a Gaussian probability distribution of unknown standard deviation S. The mean of this distribution is given by a polynomial function of the measurand Y. For the coefficients P of this polynomial a state-of-knowledge distribution is available, whereas no prior information about Y and S exists. A conditional reference prior for (Y, S) given P is derived. It shows no functional dependence on the measurand in the case that the calibration function is linear, but depends non-trivially on the measurand otherwise. This prior is compared with other priors that have been used in the literature to analyse the same calibration model. It leads to a different posterior distribution than the application of Supplement 1 to the ‘Guide to the Expression of Uncertainty in Measurement’. An example illustrates differences of results founded on the various non-informative priors.
Current recommendations for evaluating uncertainty of measurement are based on the Bayesian interpretation of probability distributions as encoding the state of knowledge about the quantities to which those distributions refer. Given a measurement model that relates an output quantity to one or more input quantities, the distribution of the former is obtained by propagating those of the latter according to the axioms of probability calculus and also, if measurement data are available, by applying Bayes' theorem.The main objective of this paper is to show that alternative ways of applying Bayes' theorem are possible, and that these alternative formulations yield the same results provided consistent use is made of measurement data and prior information. In this context the necessity of assigning non-informative priors arises often. Therefore, the second concern of the paper is to point out, by means of a specific example, that the seemingly reasonable choice of a uniform prior for a quantity about which no information is available may not conform to the accepted rules for constructing non-informative priors.
In this paper we address measurement problems involving several quantities that are interrelated by model equations. Available knowledge about some of these quantities is represented by probability density functions (PDFs), which are then propagated through the model in order to obtain the PDFs attributed to the quantities for which nothing is initially known. A formalism for analyzing such models is presented. It comprises the concept of a "base parameterization", which is used in conjunction with the change-of-variables theorem. The calculation procedure that results from this formalism is described in very general terms. Guidance is given on how to employ it in practice by presenting both an elementary example and a much more involved one.
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