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lmm comments on Open thread, 25-31 August 2014 - Less Wrong Discussion
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Can someone link to a discussion, or answer a small misconception for me?
We know P(A & B) < P(A). So if you add details to a story, it becomes less plausible. Even though people are more likely to believe it.
However, If I do an experiment, and measure something which is implied by A&B, then I would think "A&B becomes more plausible then A", Because A is more vague then A&B.
But this seems to be a contradiction.
I suppose, to me, adding more details to a story makes the story more plausible if those details imply the evidence. Sin(x) is an analytic function. If I know a complex differentiable function has roots on all multiples of pi, Saying the function is satisfied by Sin is more plausible then saying it's satisfied by some analytic function.
I think...I'm screwing up the semantics, since sin is an analytic function. But this seems to me to be missing the point.
I read a technical explanation of a technical explanation, so I know specific theories are better then vague theories (provided the evidence is specific). I guess I'm asking for clarifications on how this is formally consistent with P(A) > P(A&B).
I think we tend to intuitively "normalize" the likelihood of a complex statement. Our prior is probably Kolmogorov complexity, so if A is a 2-bit statement and B is a 3-bit statement, we would "expect" the probabilities to be P(A)=1/4, P(B)=1/8, P(A&B)=1/32. If our evidence leads us to adjust to say P(A)=1/3, P(A&B)=1/4, then while A&B is still less likely than A, there is some sense in which A&B is "higher above baseline".
Coming from the other end, predictions, this sort of makes sense. Theories that are more specific are more useful. If we have a theory that this sequence consists of odd numbers, that lets us make some prediction about the next number. If our theory is that the numbers are all primes, we can make a more specific, and therefore more useful, prediction about the next number. So even though the theory that the sequence is odd is more likely than the theory that the sequence is prime, the latter is more useful. I think that's where the idea that specific theories are better than vague theories comes from.