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taiyo comments on Book Club Update, Chapter 2 of Probability Theory - Less Wrong
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Yeah. A total derivative. The way I think about it is the dv thing there (jargon: a differential 1-form) eats a tangent vector in the y-z plane. It spits out the rate of change of the function in the direction of the vector (scaled appropriately with the magnitude of the vector). It does this by looking at the rate of change in the y-direction (the dy stuff) and in the z-direction (the dz stuff) and adding those together (since after taking derivatives, things get nice and linear).
I'm not too familiar with the functional equation business either. I'm currently trying to figure out what the heck is happening on the bottom half of page 32. Figuring out the top half took me a really long while (esp. 2.50).
I'm convinced that the inequality in eqn 2.52 shouldn't be there. In particular, when you stick in the solution S(x) = 1 - x, it's false. I can't figure out if anything below it depends on that because I don't understand much below it.
I could not figure out why alpha > 0 neither and it seems wrong to me too. But this does not look like a problem.
We know that J is an increasing function because of 2-49. So in 2-53, alpha and log(x/S(x)) must have the same sign, since the remaining of the right member tends toward 0 when q tends toward + infinity.
Then b is positive and I think it is all that matters.
However, if alpha = 0, b is not defined. But if alpha=0 then log(x/S(x))=0 as a consequence of 2-53, so x/S(x)=1. There is only one x that gives us this since S is strictly decreasing. And by continuity we can still get 2-56.
Lovely. Thanks.
I'm totally stuck on getting 2.50 from 2.48, would appreciate a hint.
K. S. Van Horn gives a few lines describing the derivation in his PT:TLoS errata. I don't understand why he does step 4 there -- it seems to me to be irrelevant. The two main facts which are needed are step 2-3 and step 5, the sum of a geometric series and the Taylor series expansion around y = S(x). Hopefully that is a good hint.
Nitpicking with his errata, 1/(1-z) = 1 + z + O(z^2) for all z is wrong since the interval of convergence for the RHS is (-1,1). This is not important to the problem since the z here will be z = exp(-q) which is less than 1 since q is positive.
It is not very important, but since you mentioned it :
The interval of convergence of the Taylor series of 1/(1-z) at z=0 is indeed (-1,1).
But "1/(1-z) = 1 + z + O(z^2) for all z" does not make sense to me.
1/(1-z) = 1 + z + O(z^2) means that there is an M such as |1/(1-z) - (1 + z)| is no greater that M*z^2 for every z close enough to 0. It is about the behavior of 1/(1-z) - (1 + z) when z tends toward 0, not when z belongs to (-1,1).
Is there anything more to getting 2.53 than just rearranging things around? I'm not sure I really understand where we get the left-hand side from.
Indeed, thanks!