Sworn testimony of the model evidence: Gaussian Mixture Importance (GAME) sampling

Elena Volpi*, Gerrit Schoups, Giovanni Firmani, Jasper A. Vrugt

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

30 Citations (Scopus)
170 Downloads (Pure)


What is the “best” model? The answer to this question lies in part in the eyes of the beholder, nevertheless a good model must blend rigorous theory with redeeming qualities such as parsimony and quality of fit. Model selection is used to make inferences, via weighted averaging, from a set of K candidate models, (Formula presented.), and help identify which model is most supported by the observed data, (Formula presented.). Here, we introduce a new and robust estimator of the model evidence, (Formula presented.), which acts as normalizing constant in the denominator of Bayes’ theorem and provides a single quantitative measure of relative support for each hypothesis that integrates model accuracy, uncertainty, and complexity. However, (Formula presented.) is analytically intractable for most practical modeling problems. Our method, coined GAussian Mixture importancE (GAME) sampling, uses bridge sampling of a mixture distribution fitted to samples of the posterior model parameter distribution derived from MCMC simulation. We benchmark the accuracy and reliability of GAME sampling by application to a diverse set of multivariate target distributions (up to 100 dimensions) with known values of (Formula presented.) and to hypothesis testing using numerical modeling of the rainfall-runoff transformation of the Leaf River watershed in Mississippi, USA. These case studies demonstrate that GAME sampling provides robust and unbiased estimates of the evidence at a relatively small computational cost outperforming commonly used estimators. The GAME sampler is implemented in the MATLAB package of DREAM and simplifies considerably scientific inquiry through hypothesis testing and model selection.

Original languageEnglish
Pages (from-to)6133-6158
Number of pages26
JournalWater Resources Research
Issue number7
Publication statusPublished - 1 Jul 2017


  • Bayesian model evidence
  • Bayesian model selection
  • bridge sampling
  • importance sampling
  • MCMC simulation


Dive into the research topics of 'Sworn testimony of the model evidence: Gaussian Mixture Importance (GAME) sampling'. Together they form a unique fingerprint.

Cite this