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Probability and Measurement Uncertainty in Physics - a Bayesian Primer

G. D'Agostini

TL;DR

Brings Bayesian probability to physics measurement, arguing that probability should reflect degree of belief about true values rather than long-run frequencies. It lays out a full framework for treating random and systematic uncertainties within a unified Bayesian model, including priors, likelihoods, and marginalization, and demonstrates with high-energy physics examples such as confidence intervals, upper limits, and unfolding. The notes contrast Bayesian and frequentist perspectives, discuss prior selection and the role of information, and provide practical guidance for applying Bayesian inference to measurement and uncertainty propagation. The result is a principled, coherent approach adopted by metrology bodies for expressing measurement uncertainty.

Abstract

Bayesian statistics is based on the subjective definition of probability as {\it ``degree of belief''} and on Bayes' theorem, the basic tool for assigning probabilities to hypotheses combining {\it a priori} judgements and experimental information. This was the original point of view of Bayes, Bernoulli, Gauss, Laplace, etc. and contrasts with later ``conventional'' (pseudo-)definitions of probabilities, which implicitly presuppose the concept of probability. These notes show that the Bayesian approach is the natural one for data analysis in the most general sense, and for assigning uncertainties to the results of physical measurements - while at the same time resolving philosophical aspects of the problems. The approach, although little known and usually misunderstood among the High Energy Physics community, has become the standard way of reasoning in several fields of research and has recently been adopted by the international metrology organizations in their recommendations for assessing measurement uncertainty. These notes describe a general model for treating uncertainties originating from random and systematic errors in a consistent way and include examples of applications of the model in High Energy Physics, e.g. ``confidence intervals'' in different contexts, upper/lower limits, treatment of ``systematic errors'', hypothesis tests and unfolding.

Probability and Measurement Uncertainty in Physics - a Bayesian Primer

TL;DR

Brings Bayesian probability to physics measurement, arguing that probability should reflect degree of belief about true values rather than long-run frequencies. It lays out a full framework for treating random and systematic uncertainties within a unified Bayesian model, including priors, likelihoods, and marginalization, and demonstrates with high-energy physics examples such as confidence intervals, upper limits, and unfolding. The notes contrast Bayesian and frequentist perspectives, discuss prior selection and the role of information, and provide practical guidance for applying Bayesian inference to measurement and uncertainty propagation. The result is a principled, coherent approach adopted by metrology bodies for expressing measurement uncertainty.

Abstract

Bayesian statistics is based on the subjective definition of probability as {\it ``degree of belief''} and on Bayes' theorem, the basic tool for assigning probabilities to hypotheses combining {\it a priori} judgements and experimental information. This was the original point of view of Bayes, Bernoulli, Gauss, Laplace, etc. and contrasts with later ``conventional'' (pseudo-)definitions of probabilities, which implicitly presuppose the concept of probability. These notes show that the Bayesian approach is the natural one for data analysis in the most general sense, and for assigning uncertainties to the results of physical measurements - while at the same time resolving philosophical aspects of the problems. The approach, although little known and usually misunderstood among the High Energy Physics community, has become the standard way of reasoning in several fields of research and has recently been adopted by the international metrology organizations in their recommendations for assessing measurement uncertainty. These notes describe a general model for treating uncertainties originating from random and systematic errors in a consistent way and include examples of applications of the model in High Energy Physics, e.g. ``confidence intervals'' in different contexts, upper/lower limits, treatment of ``systematic errors'', hypothesis tests and unfolding.

Paper Structure

This paper contains 39 sections, 43 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Probability
  3. What is probability?
  4. Subjective definition of probability
  5. Rules of probability
  6. Subjective probability and "objective"description of the physical world
  7. Conditional probability and Bayes' theorem
  8. Dependence of the probability from the status of information
  9. Conditional probability
  10. Bayes' theorem
  11. Conventional use of Bayes' theorem
  12. Bayesian statistics: learning by experience
  13. Hypothesis test (discrete case)
  14. Choice of the initial probabilities (discrete case)
  15. General criteria
  16. ...and 24 more sections

Figures (7)

  • Figure 1: Certain and uncertain events.
  • Figure 2: Venn diagrams and set properties.
  • Figure 3: Results on two physical quantities as a function of the publication date.
  • Figure 4: $R=\sigma_L/\sigma_T$ as a function of the Deep Inelastic Scattering variable $x$ as measured by experiments and as predicted by QCD.
  • Figure 6: Central limit theorem at work: the sum of $n$ variables, for two different distribution, is shown. The values of $n$ (top-down) are: 1,2,3,5,10,20,50.
  • ...and 2 more figures