Bartlett's paradox has been taken to imply that using improper priors results in Bayes factors that are not well defined, preventing model comparison in this case. We use well understood principles underlying what is already common practice, to demonstrate that this implication is not true for some improper priors, such as the Shrinkage prior due to Stein (1956). While this result would appear to expand the class of priors that may be used for computing posterior odds, we warn against the straightforward use of these priors. Highlighting the role of the prior measure in the behaviour of Bayes factors, we demonstrate pathologies in the prior measures for these improper priors. Using this discussion, we then propose a method of employing such priors by setting rules on the rate of diffusion of prior certainty.

Bayes factor, improper prior, marginal likelihood, shrinkage prior
Bayesian Analysis (jel C11), Simulation Methods; Monte Carlo Methods; Bootstrap Methods (jel C15), Time-Series Models; Dynamic Quantile Regressions (jel C32), Model Evaluation and Testing (jel C52)
Econometric Institute Research Papers
Report / Econometric Institute, Erasmus University Rotterdam
Erasmus School of Economics

Strachan, R.W, & van Dijk, H.K. (2005). Weakly informative priors and well behaved Bayes factors (No. EI 2005-40). Report / Econometric Institute, Erasmus University Rotterdam. Retrieved from