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.

A Formalism for Expressing the Probability Density Functions of Interrelated Quantities

“…It occurs when there is information on just the l quantities of a certain parameterization X b = {X b1 , K , X bl }, henceforth called the base parameterization, such that their joint PDF, denoted by f b (ξ b | I ), is either given or can be constructed by duly processing the available information (as explained, e.g., in section 3 of [1]). Here, the symbol ξ b designates the set of variables that represent the possible values of the quantities X b .…”

“…Since the information about X 1 is independent from that about X 2 , the joint PDF for these two quantities is equal to the product of their individual PDFs, f 1 It is worth mentioning that the recommendation in [3] about the use of a t-distribution follows from Bayes' theorem and the assumption that the data are generated in accordance with a Gaussian random process of unknown mean and variance. The latter then becomes a further quantity that does not enter into any parameterization because it does not appear in the measurement model.…”

“…N A PREVIOUS PAPER [1] we presented the main elements of a Bayesian procedure for assigning state of knowledge probability density functions (PDFs) to metrological quantities having essentially unique true values. In accordance with the prevalent view advocated in well-known international documents [2], [3], the mean and standard deviation of any such PDF are, respectively, taken as the best estimate and associated standard uncertainty of the quantity to which the distribution pertains, and the area under the PDF over a given interval is regarded as the probability of finding the true value of the quantity within that interval, given the available information.…”

“…The procedure explained in [1] relies on the use of Dirac delta functions for effecting the propagation of probability distributions analytically [3, subclause 5.2]. Though the delta function is a very powerful concept, the rules for its manipulation may not be familiar to some metrologists.…”

Uncertainty assessment in building energy performance with a simplified model

“…Bayesian inference about a quantity [11], is made through the probability density function that describes the information acquired from measurement and the knowledge about the quantity before the measurement is performed [16,17]. Bayes formula is a mechanism that combines the prior information on the parameters and the information provided by the measured data.…”