For Hypothesis Testing and Model Selection, the Bayesian approach is attracting considerable attention. The reasons for this attention include: i) it yields posterior probabilities of the models (and not simply accept-reject rules); ii) it is a predictive approach; and iii) it automatically incorporates the principle of scientific parsimony. Until recently, obtaining such benefits through the Bayesian approach required elicitation of proper subjective prior distributions, or the use of approximations (such as BIC) of questionable generality. In Berger and Pericchi (1996), the Intrinsic Bayes Factor Strategy was introduced, and shown to be an automatic default method corresponding to an actual (and sensible) Bayesian analysis. In particular, it was shown that the Intrinsic Bayes Factor yields an answer which is asymptotically equivalent to the use of a specific (and reasonable) proper prior distribution, called the Intrinsic Prior. Indeed, the IBF method can also be thought of as a method for constructing default proper priors appropriate for model comparisons. In this paper we study an implementation of the IBF strategy called the Median IBF. This seems to be a simple and very generally applicable IBF, which works well for nested or non-nested models, and even for small or moderate sample sizes; some of these situations can cause difficulties for other versions of IBFs. Postscript File (525kB)