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


C. Elster, G. Wübbeler  Bayesian inference using a noninformative prior for linear Gaussian random coefficient regression with inhomogeneous withinclass variances.  Comput. Stat., 32(1), 5169  2017 
S. Demeyer, N. Fischer  Bayesian framework for proficiency tests using auxiliary information on laboratories  Accreditation and Quality Assurance, February 2017, Volume 22, Issue 1, pp 1–19  2017 
O. Bodnar, A. Link and C. Elster  Objective Bayesian inference for a generalized marginal random effects model  Bayesian Analysis  2016 
A. Allard, N. Fischer, G. Ebrard, B. Hay, P. M. Harris, L. Wright, D. Rochais, J. Mattout  A multithermogram based Bayesian model for the determination of the thermal diffusivity of a material  Metrologia  2016 
C. Elster and G. Wübbeler  Bayesian regression versus application of least squares—an example  Metrologia, 53(1), S10  2016 
K. Klauenberg and C. Elster  Markov chain Monte Carlo methods: an introductory example  Metrologia, 53(1), S32  2016 
O. Bodnar, C. Elster, J. Fischer, A. Possolo and B. Toman  Evaluation of uncertainty in the adjustment of fundamental constants  Metrologia, 53(1), S46  2016 
C. Elster and G. Wübbeler  Bayesian inference using a noninformative prior for linear Gaussian random coefficient regression with inhomogeneous withinclass variances  Comput. Stat., 30(4)  2016 
M. Dierl, T. Eckhard, B. Frei, M. Klammer, S. Eichstädt and C. Elster  Improved estimation of reflectance spectra by utilizing prior knowledge  Journal of the Optical Society of America A Vol. 33, Issue 7, pp. 13701376  2016 
G. Wübbeler, O. Bodnar and C. Elster  Bayesian hypothesis testing for key comparisons  Metrologia, vol 53(4); pp 11311138  2016 
O. Bodnar, A. Link, B. Arendacká, A. Possolo, C. Elster  Bayesian estimation in random effects metaanalysis using a noninformative prior  Statistics in Medicine, 39(2), 378399  2016 
K. Klauenberg, M. Walzel, B. Ebert and C. Elster  Informative prior distributions for ELISA analyses  Biostatistics  2015 
GJP Kok, AMH van der Veen, PM Harris, IM Smith, C Elster  Bayesian analysis of a flow meter calibration problem  Metrologia 52, 392399  2015 
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. Pendrill  A Guide to Bayesian Inference for Regression Problems  Deliverable of EMRP project NEW04 “Novel mathematical and statistical approaches to uncertainty evaluation”  2015 
K. Klauenberg, G. Wübbeler, B. Mickan, P. M. Harris, and C. Elster  A Tutorial on Bayesian Normal Linear Regression  Metrologia, 52(6)  2015 
C. Elster and G. Wübbeler  Bayesian inference using a noninformative prior for linear Gaussian random coefficient regression with inhomogeneous withinclass variances  Comput. Stat. 30 (4)  2015 
C. Elster  Bayesian uncertainty analysis compared with the application of GUM and its supplements  Metrologia 51, S159S166  2014 
S. Heidenreich, H. Gross, M.A. Henn, C. Elster, and M. Bär  A surrogate model enables a Bayesian approach to the inverse problem of scatterometry  J. Phys. : Conf. Ser. 490, 012007  2014 
O. Bodnar and C. Elster  Analytical derivation of the reference prior by sequential maximization of Shannon's mutual information in the multigroup parameter case  Journal of Statistical Planning and Inference 147, 106116  2014 
O. Bodnar, A. Link and C. Elster  Bayesian treatment of a random effects model for the analysis of key comparisons  Talk at (MATHMET) International Workshop on Mathematics and Statistics for Metrology, March, 2426, 2014, Berlin  2014 
O. Bodnar, A. Link, K. Klauenberg, K. Jousten, and C. Elster  Application of Bayesian model averaging using a fixed effects model with linear drift for the analysis of key comparison CCM.PK12  Meas. Tech. 56, 584590  2013 
G. Wübbeler, F. Schmähling, J. Beyer, J. Engert, and C. Elster  Analysis of magnetic field fluctuation thermometry using Bayesian inference  Meas. Sci. Technol. 23, 125004 (9pp).  2012 
B. Toman, J. Fischer, and C. Elster  Alternative analyses of measurements of the Planck constant  Metrologia 49, 567571  2012 
C. Elster and I. Lira  On the choice of a noninformative prior for Bayesian inference of discretized normal observations  Comput. Stat. 27, 219235  2012 
K. Klauenberg and C. Elster  The multivariate normal mean  sensitivity of its objective Bayesian estimates  Metrologia 49, 395400  2012 
O. Bodnar, G. Wübbeler, and C. Elster  Comparison 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 Jersey  2012 
B. Toman, D.L. Duewer, H.G. Aragon, F.R. Guenther and G.C. Rhoderick  A Bayesian approach to the evaluation of comparisons of individually valueassigned reference materials  Analytical and Bioanalytical Chemistry 403(2), 537548  2012 
K. Klauenberg, B. Ebert, J. Voigt, M. Walzel, J. E. Noble, A. E. Knight and C. Elster  Bayesian analysis of an international ELISA comparability study  Clin. Chem. Lab. Med  2011 
S. Demeyer, J.L. Foulley, N. Fischer and G. Saporta  Bayesian analysis of structural equation models using parameter expansion  Learning and data science, L. Bottou, F. Murtagh, M. GettlerSumma, B. Goldfarb, C. Pardoux and M. Touati (eds.), Chapman & Hall  2011 
C. Elster and B. Toman  Bayesian uncertainty analysis for a regression model versus application of GUM supplement 1 to the leastsquares estimate  Metrologiam 48 (5), 233  2011 
D. Grientschnig and I. Lira  Reassessment of a calibration model by Bayesian reference analysis  Metrologia 48 (1), L7  2011 
A.Forbes, J. Alves e Sousa  The GUM, Bayesian inference and observation and measurement equations  Measurement  2011 
C. Elster and B. Toman  Analysis of key comparisons: estimating laboratories' biases by a fixed effects model using Bayesian model averaging  Metrologia 47, 113119  2010 
C. Elster and B. Toman  Bayesian uncertainty analysis under prior ignorance of the measurand versus analysis using the Supplement 1 to the Guide: a comparison  Metrologia, vol 46(3), 261266  2009 
C. Elster, W. Wöger and M. G. Cox  Draft GUM Supplement 1 and Bayesian analysis  Metrologia 44, 3132  2007 
A. Possolo and B. Toman  Assessment of measurement uncertainty via observation equations  Metrologia 44(6), 464  2007 
B. Toman  Bayesian approaches to calculating a reference value in key comparisons  Technometrics 49, 8187  2007 
B. Toman  Linear statistical models in the presence of systematic effects requiring a Type B evaluation of uncertainty  Metrologia 43(1), 27  2006 
P.H. Garthwaite, J.B. Kadane and A. O'Hagan  Statistical Methods for Eliciting Probability Distributions  Journal of the American Statistical Association 100, 680701  2005 
G.M. Rocha and G.A. Kyriazisa  A software for the evaluation of the stability of measuring standards using Bayesian statistics  In Proceedings of the 13th International Symposium on Measurements for Industry Applications, 386391  2004 
R. Kacker and A. Jones  On the use of Bayesian statistics to make the Guide to the Expression of Uncertainty in Measurement consistent  Metrologia, 40, 235248  2003 