Crutchfield, J. P.,Ellison, C. J.,Mahoney, J. R.
We show why the amount of information communicated between the past and future-the excess entropy-is not in general the amount of information stored in the present-the statistical complexity. This is a puzzle, and a long-standing one, since the former describes observed behavior, while optimal prediction requires the latter. We present a closed-form expression for the excess entropy in terms of optimal causal predictors and retrodictors-both epsilon machines of computational mechanics. This leads us to two new system invariants: causal irreversibility-the asymmetry between the causal representations-and crypticity-the degree to which a process hides its state information.