Notes

Chapter 6: Starting from Randomness

Section 7: The Notion of Attractors


Entropy estimates

Entropies h[n] computed from blocks of size n always decrease with n; the quantity n h[n] is always convex (negative second difference) with respect to n. At least at a basic level, to compute topological entropy one needs in effect to count every possible sequence that can be generated. But one can potentially get an estimate of measure entropy just by sampling possible sequences. One problem, however, is that even though such sampling may give estimates of probabilities that are unbiased (and have Gaussian errors), a direct computation of measure entropy from them will tend to give a value that is systematically too small. (A potential way around this is to use the theory of unbiased estimators for polynomials just above and below p Log[p].)


From Stephen Wolfram: A New Kind of Science [citation]