The European Centre for
Mathematics and Statistics in Metrology

Glossary: Bayesian inference

In Bayesian inference, any unknown -– observables (data) as well as unobservables (parameters and auxiliary variables) -– is considered to be random and assigned a probability distribution. These distributions best summarize the available information, and Bayesian inference then updates the knowledge on unobservables at the outset, with information on them contained in the data. Technically, this is done through Bayes’ theorem
\begin{equation*} \label{int_Bay_eq1}
\pi(\pmb{\theta}| \boldsymbol{y}) \propto \pi(\pmb{\theta}) \times l(\pmb{\theta}; \boldsymbol{y}) \,,
where $\pi(\pmb{\theta})$ is the prior distribution expressing the prior knowledge about the parameters, $l(\boldsymbol{\theta}; \boldsymbol{y})$ denotes the likelihood function,
and $\pi(\pmb{\theta}| \boldsymbol{y})$ is the posterior distribution that combines the prior belief about
$\pmb{\theta}$ with the information contained in the data.

Bayesian statistics has the advantage of being in line with future GUM revisions, it may account for available prior knowledge and is well suited for many metrological problems (such as regression). Including prior (i.e. more) information into the analysis usually implies more certainty, that is less uncertainty, it often leads to more reliable estimates, and in some cases is essential to obtain meaningful results at all. Thoughtfully specified prior information may even save money by compensating additional measurements. Another advantage of the Bayesian approach is that the information gained in one experiment can be taken into account completely in the analysis of a subsequent, related experiment. This is particularly important for reliable uncertainty propagation.

Practical challenges in the application of Bayesian methods often are the selection of a prior distribution and the computation of numerical results. For regression problems, the EMRP project NEW04 developed a guide to support metrologists to approach these challenges. A good deal of metrological research exists examining the conditions for (non-)equivalence of the Bayesian and the GUM approach applying it to diverse problems and many more.

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