# MATHMET

## 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
$$\begin{equation*} \label{int_Bay_eq1} \pi(\pmb{\theta}| \boldsymbol{y}) \propto \pi(\pmb{\theta}) \times l(\pmb{\theta}; \boldsymbol{y}) \,, \end{equation*}$$
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.

### Related literature

Authors
Title
Journal
Year
C. Elster, G. WübbelerBayesian inference using a noninformative prior for linear Gaussian random coefficient regression with inhomogeneous within-class variances.Comput. Stat., 32(1), 51--692017
S. Demeyer, N. FischerBayesian framework for proficiency tests using auxiliary information on laboratoriesAccreditation and Quality Assurance, February 2017, Volume 22, Issue 1, pp 1–192017
O. Bodnar, A. Link and C. ElsterObjective Bayesian inference for a generalized marginal random effects modelBayesian Analysis2016
A. Allard, N. Fischer, G. Ebrard, B. Hay, P. M. Harris, L. Wright, D. Rochais, J. MattoutA multi-thermogram based Bayesian model for the determination of the thermal diffusivity of a materialMetrologia2016
C. Elster and G. WübbelerBayesian regression versus application of least squares—an exampleMetrologia, 53(1), S102016
K. Klauenberg and C. ElsterMarkov chain Monte Carlo methods: an introductory exampleMetrologia, 53(1), S322016
O. Bodnar, C. Elster, J. Fischer, A. Possolo and B. TomanEvaluation of uncertainty in the adjustment of fundamental constantsMetrologia, 53(1), S462016
C. Elster and G. WübbelerBayesian inference using a noninformative prior for linear Gaussian random coefficient regression with inhomogeneous within-class variancesComput. Stat., 30(4)2016
M. Dierl, T. Eckhard, B. Frei, M. Klammer, S. Eichstädt and C. ElsterImproved estimation of reflectance spectra by utilizing prior knowledgeJournal of the Optical Society of America A Vol. 33, Issue 7, pp. 1370-13762016
G. Wübbeler, O. Bodnar and C. ElsterBayesian hypothesis testing for key comparisonsMetrologia, vol 53(4); pp 1131-11382016
O. Bodnar, A. Link, B. Arendacká, A. Possolo, C. ElsterBayesian estimation in random effects meta-analysis using a non-informative priorStatistics in Medicine, 39(2), 378--3992016
K. Klauenberg, M. Walzel, B. Ebert and C. ElsterInformative prior distributions for ELISA analysesBiostatistics2015
GJP Kok, AMH van der Veen, PM Harris, IM Smith, C ElsterBayesian analysis of a flow meter calibration problemMetrologia 52, 392-3992015
C. Elster, K. Klauenberg, M. Walzel, G. Wübbeler, P. Harris, M. Cox, C. Matthews, I. Smith, L. Wright, A. Allard, N. Fischer, S. Cowen, S. Ellison, P. Wilson, F. Pennecchi, G. Kok, A. van der Veen, and L. PendrillA Guide to Bayesian Inference for Regression ProblemsDeliverable of EMRP project NEW04 “Novel mathematical and statistical approaches to uncertainty evaluation”2015
K. Klauenberg, G. Wübbeler, B. Mickan, P. M. Harris, and C. ElsterA Tutorial on Bayesian Normal Linear RegressionMetrologia, 52(6)2015
C. Elster and G. WübbelerBayesian inference using a noninformative prior for linear Gaussian random coefficient regression with inhomogeneous within-class variancesComput. Stat. 30 (4)2015
C. ElsterBayesian uncertainty analysis compared with the application of GUM and its supplementsMetrologia 51, S159-S1662014
S. Heidenreich, H. Gross, M.-A. Henn, C. Elster, and M. Bär A surrogate model enables a Bayesian approach to the inverse problem of scatterometryJ. Phys. : Conf. Ser. 490, 0120072014
O. Bodnar and C. ElsterAnalytical derivation of the reference prior by sequential maximization of Shannon's mutual information in the multi-group parameter caseJournal of Statistical Planning and Inference 147, 106-1162014
O. Bodnar, A. Link and C. ElsterBayesian treatment of a random effects model for the analysis of key comparisonsTalk at (MATHMET) International Workshop on Mathematics and Statistics for Metrology, March, 24-26, 2014, Berlin2014
O. Bodnar, A. Link, K. Klauenberg, K. Jousten, and C. ElsterApplication of Bayesian model averaging using a fixed effects model with linear drift for the analysis of key comparison CCM.P-K12Meas. Tech. 56, 584-5902013
G. Wübbeler, F. Schmähling, J. Beyer, J. Engert, and C. ElsterAnalysis of magnetic field fluctuation thermometry using Bayesian inferenceMeas. Sci. Technol. 23, 125004 (9pp).2012
B. Toman, J. Fischer, and C. ElsterAlternative analyses of measurements of the Planck constantMetrologia 49, 567-5712012
C. Elster and I. LiraOn the choice of a noninformative prior for Bayesian inference of discretized normal observationsComput. Stat. 27, 219-2352012
K. Klauenberg and C. ElsterThe multivariate normal mean - sensitivity of its objective Bayesian estimatesMetrologia 49, 395-4002012
O. Bodnar, G. Wübbeler, and C. ElsterComparison of different choices for a prior under partial information in a bayesian analysis"Advanced Mathematical & Computational Tools in Metrology and Testing IX" , Series on Advances in Mathematics for Applied Sciences vol. 84, eds. F. Pavese, M. Bär, J.-R. Filtz, A. B. Forbes, L. Pendrill, K. Shirono. World Scientific New Jersey2012
B. Toman, D.L. Duewer, H.G. Aragon, F.R. Guenther and G.C. RhoderickA Bayesian approach to the evaluation of comparisons of individually value-assigned reference materialsAnalytical and Bioanalytical Chemistry 403(2), 537-5482012
K. Klauenberg, B. Ebert, J. Voigt, M. Walzel, J. E. Noble, A. E. Knight and C. ElsterBayesian analysis of an international ELISA comparability studyClin. Chem. Lab. Med2011
S. Demeyer, J.-L. Foulley, N. Fischer and G. SaportaBayesian analysis of structural equation models using parameter expansionLearning and data science, L. Bottou, F. Murtagh, M. Gettler-Summa, B. Goldfarb, C. Pardoux and M. Touati (eds.), Chapman & Hall2011
C. Elster and B. TomanBayesian uncertainty analysis for a regression model versus application of GUM supplement 1 to the least-squares estimateMetrologiam 48 (5), 2332011
D. Grientschnig and I. LiraReassessment of a calibration model by Bayesian reference analysisMetrologia 48 (1), L72011
A.Forbes, J. Alves e SousaThe GUM, Bayesian inference and observation and measurement equationsMeasurement2011
C. Elster and B. TomanAnalysis of key comparisons: estimating laboratories' biases by a fixed effects model using Bayesian model averagingMetrologia 47, 113-1192010
C. Elster and B. TomanBayesian uncertainty analysis under prior ignorance of the measurand versus analysis using the Supplement 1 to the Guide: a comparisonMetrologia, vol 46(3), 261-2662009
C. Elster, W. Wöger and M. G. CoxDraft GUM Supplement 1 and Bayesian analysisMetrologia 44, 31-322007
A. Possolo and B. TomanAssessment of measurement uncertainty via observation equationsMetrologia 44(6), 4642007
B. TomanBayesian approaches to calculating a reference value in key comparisonsTechnometrics 49, 81-872007
B. TomanLinear statistical models in the presence of systematic effects requiring a Type B evaluation of uncertaintyMetrologia 43(1), 272006
P.H. Garthwaite, J.B. Kadane and A. O'HaganStatistical Methods for Eliciting Probability DistributionsJournal of the American Statistical Association 100, 680-7012005
G.M. Rocha and G.A. KyriazisaA software for the evaluation of the stability of measuring standards using Bayesian statisticsIn Proceedings of the 13th International Symposium on Measurements for Industry Applications, 386-3912004
R. Kacker and A. JonesOn the use of Bayesian statistics to make the Guide to the Expression of Uncertainty in Measurement consistentMetrologia, 40, 235-2482003
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