Bayesian Methods in Measurement Uncertainty Evaluation

Bayesian methods offer a coherent framework for quantifying and propagating uncertainty by combining prior knowledge with experimental data. Unlike classical approaches, which treat uncertainty solely as a reflection of random error or adhere strictly to frequency‐based confidence intervals, the Bayesian paradigm interprets uncertainty as a degree of belief. This perspective allows metrologists to incorporate historical information, expert judgement and theoretical constraints directly into the evaluation of a measurand, and to update uncertainty estimates dynamically as new observations become available. Central to this approach are the concepts of prior and posterior distributions, which respectively capture initial beliefs about a quantity and their revision in light of measurement evidence. Bayesian computation often employs Monte Carlo techniques to characterise complex, non‐Gaussian and correlated uncertainty contributions, yielding full probability distributions for measurands rather than single‐value estimates. This richness of information supports more informed decision making in high‐stakes applications such as international trade in energy commodities, calibration of high‐precision instrumentation and safety assurance in nuclear and aerospace industries.

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Technical terms

Bayes’ theorem: A mathematical rule for updating the probability of a hypothesis by combining a prior distribution with a likelihood function derived from data.

Prior distribution: A probability distribution expressing initial beliefs about a measurand before new observations are taken.

Likelihood function: A function describing the probability of observing the experimental data for each possible value of the parameters being estimated.

Posterior distribution: The updated probability distribution for a measurand obtained by applying Bayes’ theorem to the prior and the likelihood.

Monte Carlo method: A computational technique that uses random sampling to propagate uncertainty through complex measurement models and derive probability distributions.

Coverage interval: A range within which a specified proportion of the posterior probability density for a measurand is contained, analogous to a credibility interval.

References

1. Uncertainty in discrete-time integration — The case of static gas meters. Measurement (2024).
2. Bayesian analysis of uncertainties in circle, straight-line and ellipse fitting considering a-priori knowledge − comparative analysis with total-least-squares approaches. Surface Topography Metrology and Properties (2024).
3. Simple informative prior distributions for Type A uncertainty evaluation in metrology. Metrologia (2023).