Bayes Factors, relations to Minimum Description Length, and overlapping model classes

This article presents a non-technical perspective on two prominent methods for analyzing experimental data in order to select among model classes. Each class consists of model instances; each instance predicts a unique distribution of data outcomes. One method is Bayesian Model Selection (BMS), instantiated with the Bayes factor. The other is based on the Minimum Description Length principle (MDL), instantiated by a variant of Normalized Maximum Likelihood (NML): the variant is termed NML* and takes prior probabilities into account. The methods are closely related. The Bayes factor is a ratio of two values: V₁ for model class M₁, and V₂ for M₂. Each V_jis the sum over the instances of M_j,of thejoint probabilities (prior times likelihood) for the observed data, normalized by a sum of such sums for all possible data outcomes. NML* is qualitatively similar: The value it assigns to each class is the maximum over the instances in M_iof the joint probability for the observed data normalized by a sum of such maxima for all possible data outcomes. The similarity of BMS to NML* is particularly close when model classes do not have instances that overlap, a way of comparing model classes that we advocate generally. These observations and suggestions are illustrated throughout with use of a simple example borrowed from Heck, Wagenmakers, and Morey (2015) in which the instances predict a binomial distribution of number of successes in N trials. The model classes posit the binomial probability of success to lie in various regions of the interval [0,1]. We illustrate the theory and the example not with equations but with tables coupled with simple arithmetic. Using the binomial example we carry out comparisons of BMS and NML* that do and do not involve model classes that overlap, and do and do not have uniform priors. When the classes do not overlap BMS and NML* produce qualitatively similar results.