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## Convergence of Discrete MDL for Sequential Prediction

Authors:Jan Poland and Marcus Hutter (2004) Comments:16 pages Subj-class:Probability Theory; Complexity; Learning Reference:Proceedings of the 17th Annual Conference on Learning Theory (COLT 2004) pages 300-314 Report-no:IDSIA-03-04 and cs.LG/0404057 Paper:LaTeX - PostScript - PDF - Html/Gif Slides:PowerPoint (needs tex4ppt) - PDF

Keywords:Minimum Description Length, Sequence Prediction, Convergence, Discrete Model Classes, Universal Induction, Stabilization, Algorithmic Information Theory.

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.

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@InProceedings{Hutter:04mdl2p, author = "Marcus Hutter", title = "Convergence of Discrete {MDL} for Sequential Prediction", booktitle = "Proc. 17th Annual Conf. on Learning Theory ({COLT-2003})", series = "Lecture Notes in Artificial Intelligence", editor = "J. Shawe-Taylor and Y. Singer", publisher = "Springer", address = "Berlin", pages = "300--314", year = "2004", http = "http://www.hutter1.net/ai/mdl2p.htm", url = "http://arxiv.org/abs/cs.LG/0404057", ftp = "ftp://ftp.idsia.ch/pub/techrep/IDSIA-03-04.pdf", keywords = "Minimum Description Length, Sequence Prediction, Convergence, Discrete Model Classes, Universal Induction, Stabilization, Algorithmic Information Theory.", 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.", }

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