Specified complexity allows us to measure how surprising random outcomes are, in reference to some probabilistic model. But there are other ways of measuring surprise. In Shannon’s celebrated information theory (Shannon 1948), improbability alone can be used to measure the surprise of observing a particular random outcome, using the quantity of surprisal, which is simply the negative logarithm (base 2) of the probability of observing the outcome, namely,
where x is the observed outcome and p(x) is the probability of observing it under some distribution p. Unlikely outcomes generate large surprisal values, since they are in some sense unexpected.
But let us consider a case where all events in a set of possible outcomes are equally very unlikely. (This can happen when you have an extremely large number of equally possible outcomes, so that each of them individually has a small chance of occurring.)
Under these conditions, asking “what is the probability that an unlikely event occurs?” yields the somewhat paradoxical answer that it is guaranteed to occur! Some outcome must occur, and since each of them is unlikely, an unlikely event (with large surprisal) is guaranteed to occur. Therefore, surprisal alone cannot tell us how likely we are to witness an outcome that surprises us…
How can specified complexity help?
The paper, which is mathematical in nature, ties together several existing models of specified complexity and introduces a canonical form for which objects exhibiting large specified complexity values are unlikely (surprising!) under any given distribution. Montañez builds on much previous work, fleshing out the equivalence between specified complexity testing and p-value hypothesis testing introduced by A. Milosavljević (Milosavljević 1993; Milosavljević 1995) and later William Dembski (Dembski 2005), and giving bounds on the probability of encountering large specified complexity values for existing specified complexity models. “Measuring Surprise — A Frontier of Design Theory” at Evolution News and Science Today:
A new unified model of specified complexity
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