In this paper we provide guidance on a Bayesian uncertainty evaluation for a large class of GUM measurement models covering linear and nonlinear models. Bayesian analysis takes advantage of useful prior knowledge on the measurand, which is often available from metrologist's genuine expertise and opinion, or from previous experiments and which is neither taken into account by the GUM nor by its Supplement 1. For the considered class of measurement models, we establish the equivalence with the related statistical models and derive analytical expressions of the posterior distribution for an appropriate family of prior distributions, which allows to gain insight into the result of the Bayesian uncertainty evaluation. We extend this work to the formulation of arbitrary prior distributions for the measurand and provide some guidance to set hyperparameter values within a class of priors based on elicitation techniques, so that the resulting priors reflect the prior knowledge. Posterior distributions are calculated by Markov Chain Monte Carlo (MCMC) methods. We apply the Bayesian uncertainty evaluation to the mass calibration example of Supplement 1 and compare our results with those obtained by the GUM and its Supplement 1. In order to study the impact of the choice of method for this example, we carry out a sensitivity analysis of the results with respect to the choice of prior. We show a virtually strong effect of the prior distribution which results in reduced uncertainty estimates for a small number of observations. When using noninformative priors, we obtain results comparable to those achieved by GUM-S1. Python code is made available that enables a Bayesian uncertainty evaluation also in other applications covered by the considered class of GUM measurement models.

中文翻译：

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一类 GUM 测量模型的贝叶斯不确定性评估指南

在本文中，我们提供了关于涵盖线性和非线性模型的一大类 GUM 测量模型的贝叶斯不确定性评估的指导。贝叶斯分析利用了关于被测量的有用先验知识，这些知识通常可以从计量学家的真正专业知识和意见中获得，或者从以前的实验中获得，GUM 或其补充 1 均未考虑这些知识。 对于所考虑的测量模型类别，我们建立了与相关统计模型的等价性，并推导出了适当的先验分布族的后验分布的解析表达式，从而可以深入了解贝叶斯不确定性评估的结果。我们将这项工作扩展到被测量的任意先验分布的公式化，并提供了一些指导，以根据启发技术在一类先验内设置超参数值，以便得到的先验反映先验知识。后验分布由马尔可夫链蒙特卡罗 (MCMC) 方法计算。我们将贝叶斯不确定度评估应用于补充 1 的质量校准示例，并将我们的结果与 GUM 及其补充 1 获得的结果进行比较。 为了研究此示例中方法选择的影响，我们进行了敏感性分析关于先验选择的结果。我们展示了先验分布的强大影响，这导致对少量观察的不确定性估计减少。当使用非信息先验时，我们获得的结果与 GUM-S1 取得的结果相当。提供了 Python 代码，可以在所考虑的 GUM 测量模型类别所涵盖的其他应用程序中进行贝叶斯不确定性评估。