We propose depth as a formal measure of value. From the earliest days
of information theory it has been appreciated that information per se is not
a good measure of message value. For example, a typical sequence of coin
tosses has high information content but little value; an ephemeris, giving the
positions of the moon and planets every day for a hundred years, has no more
information than the equations of motion and initial conditions from which it
was calculated, but saves its owner the effort of recalculating these positions.
The value of a message thus appears to reside not in its information (its
absolutely unpredictable parts), nor in its obvious redundancy (verbatim
repetitions, unequal digit frequencies), but rather in what might be called
its buried redundancy—parts predictable only with difficulty, things the
receiver could in principle have figured out without being told, but only at
considerable cost in money, time, or computation. In other words, the value
of a message is the amount of mathematical or other work plausibly done
by its originator, which its receiver is saved from having to repeat.
Of course, the receiver of a message does not know exactly how it orig-
inated; it might even have been produced by coin tossing. However, the
receiver of an obviously non-random message, such as the first million bits
of π, would reject this “null” hypothesis, on the grounds that it entails
nearly a million bits worth of ad-hoc assumptions, and would favor an al-
ternative hypothesis that the message originated from some mechanism for
computing pi. The plausible work involved in creating a message, then, is
the amount of work required to derive it from a hypothetical cause involving
no unnecessary, ad-hoc assumptions. It is this notion of the message value
that depth attempts to formalize.
