The main objective of this paper is to develop a criterion for model selection when there is minimal prior information. In particular, we will derive an expression for posterior odds to compare models for the minimal prior information case. In this framework, models are compared according to their relative posterior probabilities, termed the posterior odds (conditioned on prior and data-sample information). We will show that the criterion obtained here reflects a tradeoff between parsimony (i.e. parameter space dimensionality) and data fit, has desirable invariance properties, and applies to nested and nonnested model comparisons. Furthermore, in nested model comparisons, this criterion can be interpreted in terms of an adjusted likelihood ratio test in which for large data samples the significance level is a declining function of the sample size. Finally, as a practical matter the criterion obtained here is computationally no more difficult to compute than the classical F ratio, and can be calculated easily from the output of standard regression computer programs.