Simplicity, Complexity, Unexpectedness, Cognition, Probability, Information
Atypical elements are unexpected because they are simple in their class.
Cw(b|r) = log2 Nwhere N is the number of elements in class r. This is because the "world-machine" needs log2 N bits to discriminate among all elements in r which one it will present to you (for details, see the Inverted Stamp example).
C(b|r) = C(f) + C(b|r&f)With a negligible added complexity, objects in r can be ranked according to the quantitative feature f (remember that complexity only cares about the size of algorithms, not about their execution time). Object b will be one of the firsts in this ranking. We may write:
C(b|r&f) = log2 N – A(k)where k is the number of standard deviations of b in the f-distribution of r. A(k) is the logarithm of the number of objects in r that are beyond k standard deviations for f. For a Laplace-Gauss distribution, we have A(k) ≈ 0.72 k2 + log2 k + 1.33 (for k > 1). The preceding relation holds as long as A(k) < log2 N. Beyond that, the atypical object becomes unique; it is perceived as a record or an impossibility.
U(b|r) = A(k) – C(f)and (since Cw(r) = 0 if r-object are supposed to exist):
U(b) = A(k) – C(f) – C(r)The corrective terms C(f) accounts for the fact that only simple atypical features make objects rare. The Louisiane crayfish (Procambarus clarkii) is atypical because it has as many as 200 chromosomes; this fact will seem unexpected to a geneticist, for whom the feature is simple, but a layman may remain stonily indifferent as the complexity of the feature consumes most of the complexity drop due to atypicality.
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