  Deterministic chaos permits a precise notion of a "perfect measurement" as
one that, when obtained repeatedly, captures all of the information created by
the system's evolution with minimal redundancy. Finding an optimal measurement
is challenging, and has generally required intimate knowledge of the dynamics
in the few cases where it has been done. We establish an equivalence between a
perfect measurement and a variant of the information bottleneck. As a
consequence, we can employ machine learning to optimize measurement processes
that efficiently extract information from trajectory data. We obtain
approximately optimal measurements for multiple chaotic maps and lay the
necessary groundwork for efficient information extraction from general time
series.
