Previously in series: Aiming at the Target

Yesterday I spoke of how "When I think you're a powerful intelligence, and I think I know something about your preferences, then I'll predict that you'll steer reality into regions that are higher in your preference ordering."

You can quantify this, at least in theory, supposing you have (A) the agent or optimization process's preference ordering, and (B) a measure of the space of outcomes - which, for discrete outcomes in a finite space of possibilities, could just consist of counting them - then you can quantify how small a target is being hit, within how large a greater region.

Then we count the total number of states with equal or greater rank in the preference ordering to the outcome achieved, or integrate over the measure of states with equal or greater rank. Dividing this by the total size of the space gives you the relative smallness of the target - did you hit an outcome that was one in a million? One in a trillion?

Actually, most optimization processes produce "surprises" that are exponentially more improbable than this - you'd need to try far more than a trillion random reorderings of the letters in a book, to produce a play of quality equalling or exceeding Shakespeare. So we take the log base two of the reciprocal of the improbability, and that gives us optimization power in bits.

This figure - roughly, the improbability of an "equally preferred" outcome being produced by a random selection from the space (or measure on the space) - forms the foundation of my Bayesian view of intelligence, or to be precise, optimization power. It has many subtleties:

(1) The wise will recognize that we are calculating the entropy of something. We could take the figure of the relative improbability of "equally good or better" outcomes, and call this the negentropy of the system relative to a preference ordering. Unlike thermodynamic entropy, the entropy of a system relative to a preference ordering can easily decrease (that is, the negentropy can increase, that is, things can get better over time relative to a preference ordering).

Suppose e.g. that a single switch will determine whether the world is saved or destroyed, and you don't know whether the switch is set to 1 or 0. You can carry out an operation that coerces the switch to 1; ...

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