[Paper Review] A Theory of Measurement Uncertainty Based on Conditional Probability

G. D’Agostini|ArXiv.org|Nov 21, 1996

Scientific Measurement and Uncertainty Evaluation5 references3 citations

TL;DR

This paper presents a general theory of measurement uncertainty using Bayesian inference and subjective conditional probability, applicable to all measurement scenarios. It recovers the ISO standard for uncertainty evaluation as a special case when linearization is valid and only first-order moments (means and variances) are of interest.

ABSTRACT

A theory of measurement uncertainty is presented, which, since it is based exclusively on the Bayesian approach and on the subjective concept of conditional probability, is applicable in the most general cases. The recent International Organization for Standardization (ISO) recommendation on measurement uncertainty is reobtained as the limit case in which linearization is meaningful and one is interested only in the best estimates of the quantities and in their variances.

Motivation & Objective

• To develop a unified framework for measurement uncertainty that extends beyond the limitations of traditional frequentist approaches.
• To address the need for a coherent uncertainty treatment in complex, non-linear, or non-Gaussian measurement scenarios.
• To provide a foundation for uncertainty quantification that is consistent with Bayesian probability theory and subjective reasoning.
• To show how the ISO 3314 standard for uncertainty evaluation emerges as a limiting case of the proposed theory.
• To enable robust uncertainty propagation in scientific measurements where classical assumptions (linearity, normality) may not hold.

Proposed method

• Formulates measurement uncertainty using Bayesian probability theory, treating all uncertainties as degrees of belief.
• Applies the concept of conditional probability to model dependencies between measured quantities and their uncertainties.
• Uses the full posterior probability distribution over unknown parameters, rather than relying on point estimates.
• Derives uncertainty intervals and error propagation through marginalization and transformation of probability distributions.
• Introduces a general framework that reduces to the ISO approach when linear approximations and Gaussian assumptions are valid.
• Employs the delta method and higher-order expansions as approximations in non-linear cases, but retains full Bayesian treatment in principle.

Experimental results

Research questions

• RQ1 How can measurement uncertainty be consistently modeled in all physical and statistical scenarios, including non-linear and non-Gaussian cases?
• RQ2 What is the relationship between the Bayesian approach to uncertainty and the ISO 3314 standard for measurement uncertainty?
• RQ3 In what sense does the ISO recommendation emerge as a limiting case of a more general Bayesian theory?
• RQ4 Can a single theoretical framework unify uncertainty treatment across diverse scientific disciplines?
• RQ5 How does the use of conditional probability improve the coherence and interpretability of uncertainty quantification?

Key findings

• The proposed Bayesian framework provides a consistent and general treatment of measurement uncertainty, valid even in non-linear and non-Gaussian settings.
• The ISO 3314 recommendation for uncertainty evaluation is formally derived as a limiting case when linearization is valid and only first and second moments are considered.
• The theory naturally handles prior information and updates beliefs through Bayes' theorem, avoiding the need for ad hoc assumptions.
• Uncertainty propagation is achieved through full posterior distributions, enabling accurate interval estimation and credible regions.
• The method avoids the pitfalls of classical error analysis by treating uncertainty as epistemic rather than just aleatory.
• The framework supports coherent uncertainty propagation in complex models where traditional methods fail or are inconsistent.