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Exceptions to Bartlett’s Paradox


  • Rodney W. Strachan

    () (Keele University, Department of Economics)

  • Herman K. van Dijk

    () (Erasmus University Rotterdam)


A sensible Bayesian model selection or comparison strategy implies selecting the model with the highest posterior probability. While some improper priors have attractive properties such as, eg, lower frequentist risk, it is generally claimed that Bartlett’s paradox implies that using improper priors for the parameters in alternative models results in Bayes factors that are not well defined, thus preventing model comparison in this case. In this paper we demonstrate this latter result is not generally true and so expand the class of priors that may be used for computing posterior odds to include some improper priors. We give a new representation of the issue of undefined Bayes factors and, from this representation, develop classes of improper priors from which well defined Bayes factors may be derived. The approaches involve either augmenting or normalising the prior measure for the parameters. One of these classes includes the well known and commonly employed shrinkage prior. Estimation of Bayes factors is demonstrated for a simple cointegration analysis.

Suggested Citation

  • Rodney W. Strachan & Herman K. van Dijk, 2004. "Exceptions to Bartlett’s Paradox," Keele Economics Research Papers KERP 2004/03, Centre for Economic Research, Keele University.
  • Handle: RePEc:kee:kerpuk:2004/03

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    More about this item


    Improper prior; Bayes factor; marginal likelihood; shrinkage prior; measure.;

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models


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