For concreteness, let’s say that the world model requires a trillion (“N”) bits to specify, the intended head costs 10,000 bits, and the instrumental head costs 1,000 bits. If we just applied a simplicity prior directly, we expect to spend N + 1,000 bits to learn the instrumental model rather than N + 10,000 bits to learn the intended model. That’s what we want to avoid.
Not sure if I’m misunderstanding this, but it seems to me that if it takes 10,000 bits to specify the intended head and 1000 bits to specify the instrumental head, that’s because the world model—which we’re assuming is accurate—considers humans that answer a question with a truthful and correct description of reality much rarer than humans who don’t. Or at least that’s the case when it comes to the training dataset. 10,000 − 1000 equals 9,000, so in this context “much rarer” means 2^{9,000} times rarer.
However,
Now we have two priors over ways to use natural language: we can either sample the intended head at random from the simplicity prior (which we’ve said has probability 2^{-10,000} of giving correct usage), or we can sample the environment dynamics from the simplicity prior and then see how humans answer questions. If those two are equally good priors, then only 2^{-10,000} of the possible humans would have correct usage, so conditioning on agreement saves us 10,000 bits.
So if I understand correctly, the right amount of bits saved here would be 9,000.
So now we spend (N/2 + 11,000) + (N/2 − 10,000) bits altogether, for a total of N + 1,000.
Unless I made a mistake, this would mean the total is N + 2,000 - which is still more expensive than finding the instrumental head.
Not sure if I’m misunderstanding this, but it seems to me that if it takes 10,000 bits to specify the intended head and 1000 bits to specify the instrumental head, that’s because the world model—which we’re assuming is accurate—considers humans that answer a question with a truthful and correct description of reality much rarer than humans who don’t. Or at least that’s the case when it comes to the training dataset. 10,000 − 1000 equals 9,000, so in this context “much rarer” means 2^{9,000} times rarer.
However,
So if I understand correctly, the right amount of bits saved here would be 9,000.
Unless I made a mistake, this would mean the total is N + 2,000 - which is still more expensive than finding the instrumental head.