Bayesian assessment of uncertainty in metrology: A tutorial

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.