Model Selection Using Information Theory and the MDL Principle

Information theory offers a coherent, intuitive view of model selection. This perspective arises from thinking of a statistical model as a code, an algorithm for compressing data into a sequence of bits. The description length is the length of this code for the data plus the length of a description of the model itself. The length of the code for the data measures the fit of the model to the data, whereas the length of the code for the model measures its complexity. The minimum description length (MDL) principle picks the model with smallest description length, balancing fit versus complexity. Variations on MDL reproduce other well-known methods of model selection. Going further, information theory allows one to choose from among various types of models, permitting the comparison of tree-based models to regressions. A running example compares several models for the well-known Boston housing data.