Predictability , Complexity , and Learning

We deŽne 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 Žnite, grow logarithmically, or grow as a fractional power law. If the time series allows us to learn a model with a Žnite number of parameters, then Ipred(T) grows logarithmically with a coefŽcient that counts the dimensionality of the model space. In contrast, power-law growth is associated, for example, with the learning of inŽnite parameter (or nonparametric) models such as continuous functions with smoothness constraints. There are connections between the predictive information and measures of complexity that have been deŽned both in learning theory and the analysis of physical systems through statistical mechanics and dynamical systems theory. Furthermore, in the same way that entropy provides the unique measure of available information consistent with some simple and plausible conditions, we argue that the divergent part of Ipred(T) provides the unique measure for the complexity of dynamics underlying a time series. Finally, we discuss how these ideas may be useful in problems in physics, statistics, and biology.