If you search the web for `measurement uncertainty', you will obtain a list of some 14500 items. This confirms, if necessary, the pervasiveness of this concept. Curiously, I am not aware of a scientific book specifically devoted to the evaluation of the measurement uncertainty. Therefore, this book certainly deserves careful consideration. If you go through it, you quickly realize that it is a multi-faceted book. It is an introduction to metrology, especially in chapter 1, due to T J Quinn, FRS, Director of the Bureau International des Poids et Mesures. It is an introductory course on uncertainty, largely but not exclusively based on the reference document, the Guide to the Expression of Uncertainty in Measurement (GUM), issued in 1995 by the seven leading organizations involved in measurement, such as the BIPM, the IUPAC, the IUPAP and so on. It is also a rich collection of examples and sometimes of curiosities (see Peelle's Pertinent Puzzle) taken from a wide spectrum of specialized applications. Furthermore, it contains an overview of one of the developments under consideration by the Joint Committee for Guides in Metrology (the body currently in charge of the GUM); namely, the treatment of the case of more than one measurand, which gives the author an occasion for an excursus on least squares and their application in metrology. From this viewpoint, the reader will find an answer to a very large number of possible questions concerning the routine uncertainty evaluation in, say, a calibration laboratory.However, in my opinion the distinctive feature of the book is the Bayesian flavour that one can perceive here and there from the first few chapters and which really begins in chapter 6, Bayesian Inference, the longest of the book. As a matter of fact, this book is essentially a manifesto of Bayesian principles applied to measurement uncertainty, and as such the title could be somewhat misleading to the unprepared reader. The application of Bayesian techniques to measurement is largely due to German scientists, with whom the author has cooperated. This approach is attractive, in that it provides a natural way to combine fresh data from the current experiment with prior knowledge such as, for example, values coming from previous calibrations.However, Bayesian techniques are far from being universally accepted within the metrologist community. If you search the web by crossing `measurement uncertainty' with `Bayes' you get only 350 items. There are essentially two reasons for this scarce acceptance. The first has to do with a supposed amount of subjectivity unavoidable in some cases in the assignment of the prior distribution, although the Bayesian theory can provide a sufficiently convincing motivation. The second reason is more practical: a strict application of Bayesian principles leads, even for comparatively simple cases, to complicated expressions which in most cases must be solved numerically. In addition, application to the case of n repeated measurements, w...

Since the type A evaluation of standard uncertainty according to the rules given in the ISO Guide to the Expression of Uncertainty in Measurement does not take into account the uncertainty that arises from the limited resolution of indicating devices, we show in this design note that the treatment of this problem calls for a quantity about which no statistical information is available; therefore, its uncertainty has to be derived from a type B evaluation.

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.

Bayesian analysis is used to evaluate the standard uncertainty and coverage probability corresponding to a measurand Z modelled as the sum of two quantities X and Y . It is assumed that the information about X is cast in the form of Gaussian or rectangular probability density functions, whereas that about Y consists of a series of independently measured values. Prior information about Z is also taken into account. Results are compared with those obtained using the procedure recommended in the ISO Guide to the Expression of Uncertainty in Measurement. It is shown that the application of Bayes' theorem leads to an improved method for evaluation of uncertainty, particularly in the case of the coverage probability.

When constructing a coverage interval from the probability density function that describes the state of knowledge about a measurand, it seems reasonable to expect that the long-run success rate of that interval will be about equal to the stipulated coverage probability. Through a specific example, the validity of this criterion is examined.