Prediction in M-complete Problems with Limited Sample Size

Abstract

We define a new Bayesian predictor called the posterior weighted median (PWM) and compare its performance to several other predictors including the Bayes model average under squared error loss, the Barbieri-Berger median model predictor, the stacking predictor, and the model average predictor based on Akaike’s information criterion. We argue that PWM generally gives better performance than other predictors over a range of $\mathcal{M}$-complete problems. This range is between the $\mathcal{M}$-closed-$\mathcal{M}$-complete boundary and the $\mathcal{M}$-complete-$\mathcal{M}$-open boundary. Indeed, as a problem gets closer to $\mathcal{M}$-open, it seems that $\mathcal{M}$-complete predictive methods begin to break down. Our comparisons rest on extensive simulations and real data examples.

As a separate issue, we introduce the concepts of the ‘Bail out effect’ and the ‘Bail in effect’. These occur when a predictor gives not just poor results but defaults to the simplest model (‘bails out’) or to the most complex model (‘bails in’) on the model list. Either can occur in $\mathcal{M}$-complete problems when the complexity of the data generator is too high for the predictor scheme to accommodate.