• Corpus ID: 31182916

# Predictability , Complexity , and Learning

@inproceedings{US2002PredictabilityC,
title={Predictability , Complexity , and Learning},
author={A. U.S.},
year={2002}
}
• A. U.S.
• Published 2002
• Computer Science
We dene predictive information Ipred(T) as the mutual information between the past and the future of a time series. Three qualitatively different behaviors are found in the limit of large observation times T: Ipred(T) can remain nite, grow logarithmically, or grow as a fractional power law. If the time series allows us to learn a model with a nite number of parameters, then Ipred(T) grows logarithmically with a coefcient that counts the dimensionality of the model space. In contrast, power…
321 Citations

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## References

SHOWING 1-10 OF 69 REFERENCES
Information theory and learning: a physical approach
It is proved that predictive information provides the unique measure for the complexity of dynamics underlying the time series and there are classes of models characterized by {\em power-law growth of the predictive information} that are qualitatively more complex than any of the systems that have been investigated before.
Bounds for predictive errors in the statistical mechanics of supervised learning.
• Computer Science
Physical review letters
• 1995
Within a Bayesian framework, by generalizing inequalities known from statistical mechanics, general upper and lower bounds for a cumulative entropic error are calculated, which measures the success in the supervised learning of an unknown rule from examples, and find that the information gain from observing a new example decreases universally like d/m.
Information-theoretic asymptotics of Bayes methods
• Computer Science
IEEE Trans. Inf. Theory
• 1990
The authors examine the relative entropy distance D/sub n/ between the true density and the Bayesian density and show that the asymptotic distance is (d/2)(log n)+c, where d is the dimension of the parameter vector.
MUTUAL INFORMATION, METRIC ENTROPY AND CUMULATIVE RELATIVE ENTROPY RISK
• Mathematics
• 1997
Assume {P θ : θ ∈ Θ} is a set of probability distributions with a common dominating measure on a complete separable metric space Y. A state θ * ∈Θ is chosen by Nature. A statistician obtains n
Statistical Inference, Occam's Razor, and Statistical Mechanics on the Space of Probability Distributions
A precise understanding of how Occam's razor, the principle that simpler models should be preferred until the data justify more complex models, is automatically embodied by probability theory is arrived at.
Mutual Information, Fisher Information, and Population Coding
• Computer Science
Neural Computation
• 1998
It is shown that in the context of population coding, the mutual information between the activity of a large array of neurons and a stimulus to which the neurons are tuned is naturally related to the Fisher information.
Unsupervised and supervised learning: Mutual information between parameters and observations
• Computer Science
• 1999
The exact bounds and asymptotic behaviors for the mutual information as a function of the data size and of some properties of the probability of theData given the parameter are derived.
General bounds on the mutual information between a parameter and n conditionally independent observations
• Mathematics, Computer Science
COLT '95
• 1995
B bounds are given in terms of the metric and information dimensions of the parameter space with respect to the Hellinger distance and the supremum of the mutual information over choices of the prior dis tribution is bound.
Universal coding, information, prediction, and estimation
A connection between universal codes and the problems of prediction and statistical estimation is established. A known lower bound for the mean length of universal codes is sharpened and generalized,