Uncertainty Quantification of Complex System Models: Bayesian Analysis

  • Jasper A. VrugtEmail author
  • Elias C. Massoud
Reference work entry


This chapter summarizes the main elements of Bayesian probability theory to help reconcile dynamic environmental system models with observations, including prediction in space (interpolation), prediction in time (forecasting), assimilation of data, and inference of the model parameters. Special attention is given to the treatment of parameter uncertainty (first-order approximations and Bayesian intervals), the prior distribution, the formulation of the likelihood function (using first-principles), the marginal likelihood, and sampling techniques used to estimate the posterior target distribution. This includes rejection sampling, importance sampling, and recent developments in Markov chain Monte Carlo simulation to sample efficiently complex and/or high-dimensional target distributions, including limits of acceptability. We illustrate the application of Bayes’ theorem and inference using three illustrative examples involving the flow and storage of water in the surface and subsurface. At least some level of calibration of these models is required to match their output with observations of system behavior and response. Algorithmic recipes of the different methods are provided to simplify implementation and use of Bayesian analysis.


Hypothesis testing Bayesian analysis Prior distribution Likelihood function Posterior distribution Monte Carlo sampling Markov chain Monte Carlo simulation Data assimilation Hydrologic modeling 



The first author is supported by funding from the UC-Lab Fees Research Program Award 237285. The material presented in this chapter is part of the first author’s graduate course on “Merging Models and Data” (CEE-290) taught at the University of California, Irvine. An animated presentation of this material can be found online at The DREAM family of algorithms discussed in this chapter are implemented in DREAM Suite, an easy to use, plug-and-play, Windows program. This program can be found online at and simplifies considerably Bayesian analysis and its application to uncertainty quantification of mathematical models.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of California IrvineIrvineUSA
  2. 2.Department of Earth System ScienceUniversity of California IrvineIrvineUSA

Section editors and affiliations

  • Dmitri Kavetski
    • 1
  • Kuolin Hsu
    • 2
  • Yuqiong Liu
    • 3
  1. 1.School of Civil, Environmental and Mining Engineering, University of AdelaideAdelaideAustralia
  2. 2.Civil & Environmental Engineering, The Henry Samueli School of Engineering, University of CaliforniaIrvineUSA
  3. 3.NASA Goddard Space Flight CenterWashington D.C.USA

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