# Bayesian statistics and Monte Carlo methods

K. R. Koch 1
• 1 Institute for Geodesy and Geoinformation, Theoretical Geodesy Group, University of Bonn, Nussallee 17, 53115 , Bonn, Germany

## Abstract

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.

If the inline PDF is not rendering correctly, you can download the PDF file here.

• 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.

• 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.

• 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.

• 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.

• Blatter C., 1974, Analysis I, II, III, Springer, Berlin.

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

• Cox R.T., 1946, Probability, frequency and reasonable expectation, American J of Physics, 14, 1-13.

• 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.

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

• Jaynes E.T., 2003, Probability theory. The logic of science, Cambridge University Press, Cambridge.

• 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, www.bipm.org/en/publications/guides/.

• Koch K.R., 1999, Parameter Estimation and Hypothesis Testing in Linear Models, 2nd Ed. Springer, Berlin.

• 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.

• Koch K.R., 2007, Introduction to Bayesian Statistics, 2nd Ed. Springer, Berlin.

• Koch K.R., 2008, Evaluation of uncertainties in measurements by Monte Carlo simulations with an application for laserscanning, J Applied Geodesy, 2, 67-77.

• Koch K.R., 2015, Minimal detectable outliers as measures of reliability, J Geodesy, 89, 483-490.

• Koch K.R., 2017, ExpectationMaximization algorithm and its minimal detectable outliers, Studia Geophysica et Geodaetica, 61, 1-18.

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

• Koch K.R. and Brockmann J.M., 2016, Systematic effects in laser scanning and visualization by confidence regions, J Applied Geodesy, 10, 247-257.

• 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.

• Kusche J. (2003), A Monte-Carlo technique for weight estimation in satellite geodesy, J Geodesy, 76, 641-652.

• Lehmann R., 2013, On the formulation of the alternative hypothesis for geodetic outlier detection, J Geodesy, 87, 373-386.

• Leonard T. and Hsu J.S.J., 1999, Bayesian Methods. Cambridge University Press, Cambridge.

• 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.

• 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.

• 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.

• Press S.J., 1989, Bayesian Statistics: Principles, Models, and Applications, Wiley, New York.

• Rubinstein R.Y., 1981, Simulation and the Monte Carlo Method, Wiley, New York.

• Siebert B.R.L. and Sommer K.-D., 2004, Weiterentwicklung des GUM und Monte-Carlo-Techniken, tm-Technisches Messen, 71, 67-80.

• Sivia D.S., 1996, Data Analysis, a Bayesian Tutorial, Clarendon Press, Oxford.

• Xu P., 2001, Random simulation and GPS decorrelation, J Geodesy, 75, 408-423.

• Zellner A., 1971, An Introduction to Bayesian Inference in Econometrics, Wiley, New York.

OPEN ACCESS

### Journal of Geodetic Science

Journal of Geodetic Science is a peer-reviewed, electronic-only journal that publishes original, high-quality research on topics broadly related to Geodesy. The journal focuses on theoretical and application papers.

### Search   