Relevant information

Relevant information is the subset of information about a system's state that is actually needed to predict or control a specific set of essential variables. The concept is central to the Good Regulator theorem: a regulator need not model the entire system, but only the information relevant to the variables it must regulate. This is not merely a computational convenience but a structural feature of effective control: regulators that track irrelevant information waste resources and may overfit to noise.

The quantification of relevant information uses the information bottleneck framework: given an input variable X and a target variable Y, the relevant information is the mutual information that X carries about Y. A regulator that preserves this information while discarding the rest is, in the formal sense, optimal. The concept connects to sufficient statistics in classical statistics, where a statistic is sufficient if it preserves all relevant information for a parameter of interest.

In complex systems, the problem of identifying relevant information becomes difficult because the relevant variables may not be known in advance and may change as the system evolves. This is the adaptive regulator problem: the regulator must discover what is relevant while it is already regulating, and the cost of discovery is measured in regulation failures. The concept of relevant change — change that alters the relevant information structure — is the frontier of adaptive control theory.