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IlyaShpitser 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).
If A,B,C are binary, values of A and B are drawn from independent fair coins, and C = A XOR B, then measuring C = 1 constrains A,B to be either { 1, 1 } or { 0, 0 }, but does not constrain A alone at all.
Before we conditioned on C=1, all values of the joint variable A,B had probabilities 0.25, and all values of a single variable A had probabilities 0.5. After we conditioned on C=1, values { 0, 0 } and { 1, 1 } of A,B assume probabilities 0.5, and values { 0, 1 } and { 1, 0 } of A,B assume probabilities 0, values of a single variable A remain at probability 0.5.
By conditioning on C=1, you learn more about the joint variable A,B than about a single variable A (because your posterior for A,B changed, but your posterior for A did not), but that is not the same thing as the joint variable A,B being more plausible than the single variable A. In fact, it is still the case that p(A & B | C) <= p(A | C) for all values of A,B.
edit: others below said the same, and often better.