Entropic Strict Minimum Message Length and Its Connections to PAC-Bayes and NML

Abstract

We introduce entropic strict minimum message length (SMML), a risk-sensitive generalization of strict minimum message length coding. The proposed criterion replaces expected two-part codelength under the prior predictive distribution with an exponential certainty equivalent, thereby defining a one-parameter family of coding rules that interpolates between Bayesian average-case coding and worst-case minimax coding. We show that ordinary SMML is recovered in the risk-neutral limit, while the extreme risk-sensitive limit yields a minimax codelength criterion; when centered by the oracle maximum likelihood codelength, this criterion coincides with the normalized maximum likelihood (NML) minimax-regret principle. We further prove that entropic SMML admits a variational characterization as a Kullback--Leibler-regularized worst-case expected codelength, giving it a PAC--Bayes-type interpretation. We establish a joint asymptotic theory linking the sample size $n$ and the risk parameter $τ$, showing that in regular parametric models the transition between Bayesian, robust, and minimax coding regimes occurs on a logarithmic scale. For regular exponential families, the fixed-codebook partition remains affine in sufficient-statistic space, while the codepoints satisfy a tilted moment-matching condition and admit an interpretation as tilted Bregman centroids. These results position entropic SMML as an information-theoretic bridge between MML, PAC--Bayes, and MDL.