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In order to capture your intuition that a random sequence is "unsurprising", you want the predictor to output a distribution over {0,1} — or equivalently, a subjective probability p of the next bit being 1. The predictor tries to maximize the expectation of a proper scoring rule. In that case, the maximally unexpected sequence will be random, and the probability of the sequence will approach 2^{-n}.
Allowing the predictor to output {0, 1, ?} is kind of like restricting its outputs to {0%, 50%, 100%}.
In a random sequence, AIXI would guess on average half of the bits. My goal was to create a specific sequence, where it couldn't guess any. Not just a random sequence, but specifically... uhm... "anti-inductive"? The exact opposite of lawful, where random is merely halfway opposed. I don't care about other possible predictors, only about AIXI.
Imagine playing rock-paper-scissors against someone who beats you all the time, whatever you do. That's worse than random. This sequence would bring the mig... (read more)