Applied statistical modelers frequently have to compare models of rather different forms. To the extent that objective criteria are used to facilitate such comparisons, Akaike’s minimum AIC criterion seems to be the one most widely used, due in part, perhaps, to its ease of use and its impressive successes in some industrial applications. A coherent theory to motivate MAIC’s use with non-nested model comparisons has been lacking, however, and the present paper seeks to describe one. Not surprisingly, Akaike’s non-operative Entropy Maximization Principle turns out to provide a model of what successful performance might mean in some subtle situations involving incorrect models. This paper summarizes some new results concerning this principle, a linear stochastic regression version of Akaike’s criterion, and the related criteria of Schwarz and Hannan and Quinn. Some analyses related to a successful ship autopilot design project are presented to illustrate the application of MAIC. Our theoretical results are directed towards analyzing the performance of the model selection criteria in some general situations, including three in which the preferred model, or lack of one, seems obvious a priori. Loosely described, these three situations are:

1. The best model from one class fits the data better than all models from the other class.
2. Both model classes include the correct model, but one class has fewer parameters to be estimated than the other.
3. The two model classes have the same number of parameters to be estimated and both include the correct model. We also analyze the performance of the various criteria in two situations in which the principle of parsimony is contradicted.