General solutions for information-based and Bayesian approaches to model selection in linear regression and their equivalence

Abstract

In the paper we consider the problem of model selection for linear regression within Bayesian and information-based frameworks. For both cases we generalize known approaches (evidence-based and Akaike information criterion) and derive criterion functions in terms of (in general case non-factorial) weight priors which are assumed to be Gaussian. Optimization of these criterion functions leads to two semidefinite optimization problems which can be solved analytically. We present a method that finds best priors in both approaches and show their equivalence. Surprisingly it appears that optimal prior has rank one covariance matrix. We derive explicit condition of degenerative decision rule, i.e., regression with all weights equal to zero. We conclude with experiments that show that the proposed approach significantly reduces the time needed for model selection in comparison with alternatives based on cross-validation and iterative evidence maximization while keeping generalization ability