A comparative study on model selection and multiple model fusion

There exist quite a few criteria for penalty-based model selection. Although they have various justifications for large sample problems, their performance under small or moderate sample size is unclear which hinders the development of model combination methods using the appropriate penalty term. In this paper, we assess the performance of seven model selection criteria based on linear regression models with unknown noise variance. We set the true data generation mechanism to be within the model set as well as outside the model set. In the latter case, soft model selection through multiple model fusion is proposed and its difference from Bayesian model averaging is highlighted. The penalty term used in each model selection criterion provides a natural link to estimate the model probability without assuming any prior knowledge of the unknown parameter. An important question is whether the estimated model probabilities are consistent when multiple models are fused for prediction or interpolation. We argue that strong consistency only holds under large sample regime while soft model selection can still be better than choosing a single model with small sample size. Our numerical results using different model selection criteria for polynomial fitting indicate that the conditional model estimator (CME) has the best performance in selecting the correct model order and fusing multiple models for prediction and interpolation. The minimum description length (MDL) based criteria are next to CME and outperform Bayesian information criterion (BIC) and Akaike information criterion (AIC) significantly.