COLT '90 Proceedings of the third annual workshop on Computational learning theory
An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
Selective Sampling Using the Query by Committee Algorithm
Machine Learning
Potential-Based Algorithms in On-Line Prediction and Game Theory
Machine Learning
Universal Artificial Intelligence: Sequential Decisions Based On Algorithmic Probability
Universal Artificial Intelligence: Sequential Decisions Based On Algorithmic Probability
Adaptive Online Prediction by Following the Perturbed Leader
The Journal of Machine Learning Research
PAC-Bayes risk bounds for sample-compressed Gibbs classifiers
ICML '05 Proceedings of the 22nd international conference on Machine learning
MDL convergence speed for Bernoulli sequences
Statistics and Computing
Prediction, Learning, and Games
Prediction, Learning, and Games
The weighted majority algorithm
SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science
Fisher information and stochastic complexity
IEEE Transactions on Information Theory
The minimum description length principle in coding and modeling
IEEE Transactions on Information Theory
Information-theoretic asymptotics of Bayes methods
IEEE Transactions on Information Theory
Complexity-based induction systems: Comparisons and convergence theorems
IEEE Transactions on Information Theory
Minimum complexity density estimation
IEEE Transactions on Information Theory
Asymptotics of discrete MDL for online prediction
IEEE Transactions on Information Theory
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Bayes' rule specifies how to obtain a posterior from a class of hypotheses endowed with a prior and the observed data. There are three fundamental ways to use this posterior for predicting the future: marginalization (integration over the hypotheses w.r.t. the posterior), MAP (taking the a posteriori most probable hypothesis), and stochastic model selection (selecting a hypothesis at random according to the posterior distribution). If the hypothesis class is countable, and contains the data generating distribution (this is termed the ''realizable case''), strong consistency theorems are known for the former two methods in a sequential prediction framework, asserting almost sure convergence of the predictions to the truth as well as loss bounds. We prove corresponding results for stochastic model selection, for both discrete and continuous observation spaces. As a main technical tool, we will use the concept of a potential: this quantity, which is always positive, measures the total possible amount of future prediction errors. Precisely, in each time step, the expected potential decrease upper bounds the expected error. We introduce the entropy potential of a hypothesis class as its worst-case entropy, with regard to the true distribution. Our results are proven within a general stochastic online prediction framework, that comprises both online classification and prediction of non-i.i.d. sequences.