  We show that the way in which the Shannon entropy of sequences produced by an
information source converges to the source's entropy rate can be used to
monitor how an intelligent agent builds and effectively uses a predictive model
of its environment. We introduce natural measures of the environment's apparent
memory and the amounts of information that must be (i) extracted from
observations for an agent to synchronize to the environment and (ii) stored by
an agent for optimal prediction. If structural properties are ignored, the
missed regularities are converted to apparent randomness. Conversely, using
representations that assume too much memory results in false predictability.
