A simple method for Bayesian uncertainty evaluation in linear models

Abstract

The Guide to the Expression of Uncertainty in Measurement (GUM) has led to a harmonization of uncertainty evaluation throughout metrology. The simplicity of its employed methodology has fostered the broad acceptance of the GUM among metrologists. However, this simplicity also compromises best practice and does not provide state-of-the-art data analysis. Specifically, metrologists often possess useful prior knowledge about the measurand which cannot be accounted for by the GUM. Bayesian uncertainty evaluation, on the other hand, takes prior knowledge about the measurand as its starting point. While this technique has been successfully applied in several instances in metrology, its broad access for metrologists is still lacking. One reason for this is that application of Bayesian methods and their computation, e.g. by means of Markov chain Monte Carlo (MCMC) methods, usually requires statistical skills.In this work, we propose a simple method for Bayesian uncertainty evaluation applicable to measurement models that depend linearly on a single input quantity for which Type A information is available, and several input quantities for which Type B information is given. This category of measurement models covers many uncertainty evaluations in metrology. The approach utilizes a specific class of prior distributions to encode the prior information about the measurand. Explicit guidance is provided on how to choose the parameters of the prior in the light of one's prior knowledge. MCMC methods are not required for the calculation of results; instead, a simple Monte Carlo procedure is developed that allows to draw uncorrelated samples from the posterior distribution, which greatly simplifies convergence assessments. The proposed Bayesian approach allows the treatment of small numbers of observations for a Type A uncertainty evaluation, including the case of only a single observation. Examples are provided that illustrate the proposed approach and corresponding open source Python software is made available.

A practical two-step procedure for taking into account all available information (prior and current) about influence quantities in measurement uncertainty analysis

“…The GUM-S1 [15], which is based on the principle of propagation of distributions, has the same limitation. Wubbeler et al [28] pointed out, "… when applying the GUM or GUM-S1, … the analyst is forced to choose between the two sources of information [prior and current] when assigning an uncertainty associated with the mean of the observations. "…”

“…In general, the subjective Bayesian approach is associated with complex formulations and calculations, and requires sensitivity analysis to ensure that the results are not out of reasonable range. For example, Wubbeler et al [28] used the subjective Bayesian approach with a hypothetical informative prior for the mass calibration example of [4]. They conducted a sensitivity analysis by changing the values of the prior parameters.…”

“…They conducted a sensitivity analysis by changing the values of the prior parameters. Wubbeler et al [28] stated, "An important aspect in a Bayesian inference is the analysis of its sensitivity with respect to a variation of the prior. "…”

“…Wubbeler et al [28] recently visited this mass calibration. They used a subjective Bayesian approach with a normal inverse Gamma prior for the measurand 𝑚 𝑋 to generate the posterior distribution of 𝑚 𝑋 .…”

“…Note that EALCO [4] does not provide any prior information for 𝑚 𝑋 . Therefore, the prior used in [28] is a hypothetical informative prior, not a genuine prior.…”