NML, Bayes and true distributions

We review the normalized maximum likelihood (NML) criterion for selecting among competing models. NML is generally justified on information-theoretic grounds, via the principle of minimum description length (MDL), in a derivation that “does not assume the existence of a true, data-generating distribution.” Since this “agnostic” claim has been a source of some recent confusion in the psychological literature, we explain in detail what is meant by this statement. In doing so we discuss the work presented by Karabatsos and Walker (2006), who propose an alternative Bayesian decision-theoretic characterization of NML, which leads them to conclude that the claim of agnosticity is meaningless. In the KW derivation, one part of the NML criterion (the likelihood term) arises from placing a Dirichlet process prior over possible data-generating distributions, and the other part (the complexity term) is folded into a loss function. Whereas in the original derivations of NML, the complexity term arises naturally, in the KW derivation its mathematical form is taken for granted and not explained any further. We argue that for this reason, the KW characterization is incomplete; relatedly, we question the relevance of the characterization and we argue that their main conclusion about agnosticity does not follow.