Convergence of Discrete MDL for Sequential Prediction

TL;DR

This paper analyzes the convergence properties of the Minimum Description Length principle in sequential prediction, demonstrating its theoretical behavior and comparing it to Bayesian methods, with implications for model stability and randomness.

Contribution

It provides convergence theorems for MDL in sequence prediction, compares MDL and Bayesian methods, and explores conditions for stable MDL predictions.

Findings

MDL convergence similar to Solomonoff's theorem

Convergence bounds are exponentially larger than Bayesian mixtures

Stable MDL predictions require specific conditions

Abstract

We study the properties of the Minimum Description Length principle for sequence prediction, considering a two-part MDL estimator which is chosen from a countable class of models. This applies in particular to the important case of universal sequence prediction, where the model class corresponds to all algorithms for some fixed universal Turing machine (this correspondence is by enumerable semimeasures, hence the resulting models are stochastic). We prove convergence theorems similar to Solomonoff's theorem of universal induction, which also holds for general Bayes mixtures. The bound characterizing the convergence speed for MDL predictions is exponentially larger as compared to Bayes mixtures. We observe that there are at least three different ways of using MDL for prediction. One of these has worse prediction properties, for which predictions only converge if the MDL estimator stabilizes. We establish sufficient conditions for this to occur. Finally, some immediate consequences for complexity relations and randomness criteria are proven.