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Mathematical Modelling and Data Analysis

Department 8.4

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Title: Markov chain Monte Carlo methods: an introductory example
Author(s): K. Klauenberg and C. Elster
Journal: Metrologia
Year: 2016
Volume: 53
Issue: 1
Pages: S32
DOI: 10.1088/0026-1394/53/1/S32
Web URL: http://stacks.iop.org/0026-1394/53/i=1/a=S32
Keywords: Bayesian, MCMC, Markov chain Monte Carlo
Tags: 8.42, Unsicherheit, Regression
Abstract: When the Guide to the Expression of Uncertainty in Measurement (GUM) and methods from its supplements are not applicable, the Bayesian approach may be a valid and welcome alternative. Evaluating the posterior distribution, estimates or uncertainties involved in Bayesian inferences often requires numerical methods to avoid high-dimensional integrations. Markov chain Monte Carlo (MCMC) sampling is such a method---powerful, flexible and widely applied. Here, a concise introduction is given, illustrated by a simple, typical example from metrology. The Metropolis--Hastings algorithm is the most basic and yet flexible MCMC method. Its underlying concepts are explained and the algorithm is given step by step. The few lines of software code required for its implementation invite interested readers to get started. Diagnostics to evaluate the performance and common algorithmic choices are illustrated to calibrate the Metropolis--Hastings algorithm for efficiency. Routine application of MCMC algorithms may be hindered currently by the difficulty to assess the convergence of MCMC output and thus to assure the validity of results. An example points to the importance of convergence and initiates discussion about advantages as well as areas of research. Available software tools are mentioned throughout.

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