Simplicity Theory

Simplicity, Complexity, Unexpectedness, Cognition, Probability, Information

by Jean-Louis Dessalles
(created 2008.12.31)

(updated 2010.02.11)

Example: Inverted stamps (rarity)

Rare objects are simple and therefore unexpected

This 24-cent airmail stamp of 1918, which was erroneously printed with an inverted centre, is worth
about $200,000, about two thousand times the price of a regular copy of the same stamp

(by permission of the Smithsonian National Postal Museum,
www.postalmuseum.si.edu/exhibits/2f1a_inverts.html )

How much is this stamp unexpected? By definition, unexpectedness is the difference between generation complexity and description complexity: C_w – C. Let’s compute both terms.

Generation complexity C_w

Let’s call s the following situation: “I can see this stamp in front of me now”. What is the complexity of generating s? Suppose you consider this stamp s as member of a reference class r (r may be the class of all stamps; or the class of US-Postage 24-cent stamps). Let’s call f the feature (here the inverted image) that makes the situation remarkable. Generation complexity may be computed through the computation sequence r*d*f*s, where d is a selection operation, i.e. the fact that the actual stamp is picked out from a class:

C_w(r*d*f*s) = C_w(r) + C_w(d|r) + C_w(f|r&d) + C_w(s|r&d&f)

As far as the reference class r is supposed to cover existing objects, C_w(r) = 0. If we are ignorant of how the actual object was selected, we assume a lottery device, so that C_w(d|r) = log₂ N, where N is the number of elements in class r. Now C_w(f|r&d) = 0, as selection d is powerful enough to produce an object with property f.

C_w(r*d*f*s) = log₂ N + C_w(s|r&d&f)        (1)

The last term C_w(s|r&d&f) captures the complexity of everything in s that is not related to the object itself (where and when I saw the stamp, ...).

Description complexity C

The feature f may help discriminate s in class r, which means that description complexity may be computed through the computation sequence r*f*d*s:

C(r*f*d*s) = C(r) + C(f|r) + C(d|r&f) + C(s|r&f&d)         (2)

C(r) + C(f|r) are non-zero terms which measure the conceptual complexity of r and of f in the context of r. If there are P inverted stamps in r, then to discriminate among them on the basis of this sole feature, one needs:

C(d|r&f) = log₂ P

Rarity

If we name rarity the contribution of an infrequent object to unexpectedness:

R = U(s) – U(s|r&f&d)

Then, by subtracting (2) from (1):

R = log₂ N – log₂ P – C(f|r) – C(r)

Remark 1: Reference class r should be chosen so as to make unexpectedness maximum. Choosing a more specific class (e.g. ‘US-Postage 24-cent stamps’ instead of merely ‘stamps’) diminishes N and increases C(r), but diminishes P.

Remark 2: The choice of r is not without consequences on the computation. You may not have come upon the stamp by pure chance. Seeing an inverted stamp in an exhibition is indeed far less impressive than finding it by chance an on a letter. This means that in the former case, if r is chosen as the class of all stamps, then the assumption C_w(d|r) = log₂ N would be incorrect. Nevertheless, the above definition of R can be used by philatelists to assess the rarity of stamps. It can be also used by exhibition visitors as a quantitative feature to measure the atypicality of the various items.

Remark 3: This example is interesting also because there is an alternative computation for C_w(s), going through the sequence r*f*d*s. Complexity C_w(f|r) may be assessed through a causal story (a scenario leading to an error made by a print worker). Experts would know that C_w(f|r) amounts to log₂ N/P by using the frequency of such events. Then C_w(d|r&f) = log₂ P. The result may be identical to the above one for experts, but not for the layperson who may find the causal story highly complex (see the example on the running nuns).

Probability

The subjective probability attached to rarity, as given by formula p=2^–U, amount to (as far as R > 1):

p = P/N x 2^C(f)

Note that the first factor P/N is the classical probability value. The corrective term 2^C(f) accounts for the fact that only simple features make stamps rare. No wonder that in T. K. Tapling’s great stamp collection (XIX° century) to be seen in the British Museum in London, the nine most valuable rarities include:
- an Indian stamp of 1854 has an inverted head
- a Spanish stamp of 1851 has the wrong colour
- an Australian stamp of 1854 has an inverted frame
All these erroneous characteristics share the property of being remarkably simple.

All objects in the universe are unique. For classical probability theory, their probability is zero. Most objects are, however, unique for complex reasons. Their subjective probability is thus rather close to one for a human observer. Genuinely rare objects must be easy to describe within their class.

Bibliography

Dessalles, J-L. (2007). Complexité cognitive appliquée à la modélisation de l'intérêt narratif. Intellectica, 45 (1), 145-165.

Dessalles, J-L. (2007). Spontaneous assessment of complexity in the selection of events. Technical Report ParisTech-ENST 2007D011.

Dessalles, J-L. (2008). La pertinence et ses origines cognitives - Nouvelles théories. Paris: Hermes-Science Publications.

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