Bayesian assessment of uncertainty in metrology: a tutorial
- 11 March 2010
- journal article
- review article
- Published by IOP Publishing in Metrologia
- Vol. 47 (3), R1-R14
- https://doi.org/10.1088/0026-1394/47/3/r01
Abstract
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.Keywords
This publication has 44 references indexed in Scilit:
- A Bayesian approach to information fusion for evaluating the measurement uncertaintyRobotics and Autonomous Systems, 2009
- The generalized maximum entropy trapezoidal probability density functionMetrologia, 2008
- Uncertainty of functionals of calibration curvesMetrologia, 2007
- Measurement of radioactive samples: application of the Bayesian statistical decision theoryMetrologia, 2005
- Coverage intervals and statistical coverage intervalsMetrologia, 2004
- A Bayesian approach to the consumer's and producer's risks in measurementMetrologia, 1999
- Bayesian inference from measurement informationMetrologia, 1999
- The Selection of Prior Distributions by Formal RulesJournal of the American Statistical Association, 1996
- The Selection of Prior Distributions by Formal RulesJournal of the American Statistical Association, 1996
- Information Theory and Statistical MechanicsPhysical Review B, 1957