Skip to content
BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access March 2, 2018

Bayesian statistics and Monte Carlo methods

  • K. R. Koch EMAIL logo


The Bayesian approach allows an intuitive way to derive the methods of statistics. Probability is defined as a measure of the plausibility of statements or propositions. Three rules are sufficient to obtain the laws of probability. If the statements refer to the numerical values of variables, the so-called random variables, univariate and multivariate distributions follow. They lead to the point estimation by which unknown quantities, i.e. unknown parameters, are computed from measurements. The unknown parameters are random variables, they are fixed quantities in traditional statistics which is not founded on Bayes’ theorem. Bayesian statistics therefore recommends itself for Monte Carlo methods, which generate random variates from given distributions. Monte Carlo methods, of course, can also be applied in traditional statistics. The unknown parameters, are introduced as functions of the measurements, and the Monte Carlo methods give the covariance matrix and the expectation of these functions. A confidence region is derived where the unknown parameters are situated with a given probability. Following a method of traditional statistics, hypotheses are tested by determining whether a value for an unknown parameter lies inside or outside the confidence region. The error propagation of a random vector by the Monte Carlo methods is presented as an application. If the random vector results from a nonlinearly transformed vector, its covariance matrix and its expectation follow from the Monte Carlo estimate. This saves a considerable amount of derivatives to be computed, and errors of the linearization are avoided. The Monte Carlo method is therefore efficient. If the functions of the measurements are given by a sum of two or more random vectors with different multivariate distributions, the resulting distribution is generally not known. TheMonte Carlo methods are then needed to obtain the covariance matrix and the expectation of the sum.


Acko B. and Godina A., 2005, Verification of the conventional measuring uncertainty evaluation model with Monte Carlo simulation, Int J Simul Model, 4, 76-84.10.2507/IJSIMM04(2)3.039Search in Google Scholar

Alkhatib H. and Kutterer H., 2013, Estimation of measurement uncertainty of kinematic TLS observation process by means of Monte- Carlo methods, J Applied Geodesy, 7, 125-133.10.1515/jag-2013-0044Search in Google Scholar

Alkhatib H. and Schuh W.-D., 2007, Integration of the Monte Carlo covariance estimation strategy into tailored solution procedures for large-scale least squares problems, J Geodesy, 81, 53-66.10.1007/s00190-006-0034-zSearch in Google Scholar

Alkhatib H., Neumann I. and Kutterer H., 2009, Uncertainty modeling of random and systematic errors by means of Monte Carlo and fuzzy techniques, J Applied Geodesy, 3, 67-79.10.1515/JAG.2009.008Search in Google Scholar

Blatter C., 1974, Analysis I, II, III, Springer, Berlin. 10.1007/978-3-642-96231-8Search in Google Scholar

Box G.E.P. and Tiao G.C., 1973, Bayesian Inference in Statistical Analysis, Addison-Wesley, Reading.Search in Google Scholar

Cox R.T., 1946, Probability, frequency and reasonable expectation, American J of Physics, 14, 1-13.10.1119/1.1990764Search in Google Scholar

Gundlich B. and Kusche J., 2008, Monte Carlo integration for quasi-linear models. In Peiliang Xu, Jingnan Liu, and Athanasios Dermanis, editors, VI Hotine-Marussi Symposium on Theoretical and Computational Geodesy, p. 337-344, Springer, Berlin, Heidelberg.10.1007/978-3-540-74584-6_55Search in Google Scholar

Hennes H., 2007, Konkurrierende Genauigkeitsmaße - Potential und Schwächen aus der Sicht des Anwenders, Allgemeine Vermessungs-Nachrichten, 114, 136-146.Search in Google Scholar

Jaynes E.T., 2003, Probability theory. The logic of science, Cambridge University Press, Cambridge.10.1017/CBO9780511790423Search in Google Scholar

JCGM, 2011, Evaluation of measurement data-Supplement 2 to the “Guide to the expression of uncertainty in measurement”-Extension to any number of output quantities, JCGM 102:2011. Joint Committee for Guides in Metrology, in Google Scholar

Koch K.R., 1999, Parameter Estimation and Hypothesis Testing in Linear Models, 2nd Ed. Springer, Berlin.10.1007/978-3-662-03976-2Search in Google Scholar

Koch K.R., 2005, Determining the maximum degree of harmonic coeflcients in geopotential models by Monte Carlo methods, Studia Geophysica et Geodaetica, 49, 259-275.10.1007/s11200-005-0009-1Search in Google Scholar

Koch K.R., 2007, Introduction to Bayesian Statistics, 2nd Ed. Springer, Berlin.Search in Google Scholar

Koch K.R., 2008, Evaluation of uncertainties in measurements by Monte Carlo simulations with an application for laserscanning, J Applied Geodesy, 2, 67-77.10.1515/JAG.2008.008Search in Google Scholar

Koch K.R., 2015, Minimal detectable outliers as measures of reliability, J Geodesy, 89, 483-490.10.1007/s00190-015-0793-5Search in Google Scholar

Koch K.R., 2017, ExpectationMaximization algorithm and its minimal detectable outliers, Studia Geophysica et Geodaetica, 61, 1-18.10.1007/s11200-016-0617-ySearch in Google Scholar

Koch K.R., 2018, Monte Carlo methods. GEM-International Journal on Geomathematics, 9(1), DOI 10.1007/s13137-017-0101-z.10.1007/s13137-017-0101-zSearch in Google Scholar

Koch K.R. and Brockmann J.M., 2016, Systematic effects in laser scanning and visualization by confidence regions, J Applied Geodesy, 10, 247-257.10.1515/jag-2016-0012Search in Google Scholar

Koch K.R. and Kargoll B., 2015, Outlier detection by the EM algorithm for laser scanning in rectangular and polar coordinate systems, J Applied Geodesy, 9, 162-173.10.1515/jag-2015-0004Search in Google Scholar

Kusche J. (2003), A Monte-Carlo technique for weight estimation in satellite geodesy, J Geodesy, 76, 641-652.10.1007/s00190-002-0302-5Search in Google Scholar

Lehmann R., 2013, On the formulation of the alternative hypothesis for geodetic outlier detection, J Geodesy, 87, 373-386. 10.1007/s00190-012-0607-ySearch in Google Scholar

Leonard T. and Hsu J.S.J., 1999, Bayesian Methods. Cambridge University Press, Cambridge.Search in Google Scholar

Loredo T. J., 1990, From Laplace to Supernova SN 1987a: Bayesian inference in astrophysics, In P. F. Fougère, editor, Maximum Entropy and Bayesian Methods, p. 81-142, Dordrecht, Kluwer Academic Publ.10.1007/978-94-009-0683-9_6Search in Google Scholar

Nowel K., 2016, Application of Monte Carlo method to statistical testing in deformation analysis based on robust M-estimation, Survey Review, 48 (348), 212-223.10.1179/1752270615Y.0000000026Search in Google Scholar

Pany A., Böhm J., MacMillan D., Schuh H., Nilsson T. and Wresnik J., 2011, Monte Carlo simulations of the impact of troposphere, clock and measurement errors on the repeatability of VLBI positions, J Geodesy, 85, 39-50.10.1007/s00190-010-0415-1Search in Google Scholar

Press S.J., 1989, Bayesian Statistics: Principles, Models, and Applications, Wiley, New York.Search in Google Scholar

Rubinstein R.Y., 1981, Simulation and the Monte Carlo Method, Wiley, New York.10.1002/9780470316511Search in Google Scholar

Siebert B.R.L. and Sommer K.-D., 2004, Weiterentwicklung des GUM und Monte-Carlo-Techniken, tm-Technisches Messen, 71, 67-80.10.1524/teme. in Google Scholar

Sivia D.S., 1996, Data Analysis, a Bayesian Tutorial, Clarendon Press, Oxford.Search in Google Scholar

Xu P., 2001, Random simulation and GPS decorrelation, J Geodesy, 75, 408-423.10.1007/s001900100192Search in Google Scholar

Zellner A., 1971, An Introduction to Bayesian Inference in Econometrics, Wiley, New York.Search in Google Scholar

Received: 2017-09-27
Accepted: 2018-01-29
Published Online: 2018-03-02

© by K. R. Koch

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Downloaded on 27.9.2023 from
Scroll to top button