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On reddit.com/r/math it is possible to write latex equations by enclosing them in bracket-semicolon / semicolon-bracket delimiters. The text doesn't get modified by the server at all, but browser extensions such as these two will render the latex automatically: Firefox: http://thewe.net/tex/ Chrome: http://bit.ly/dwdPPU I tried asking a notation-heavy question here using that approach, and the latex rendered fine in the preview, but after I posted the question the site itself did something weird to the text so that it came out completely illegible. Original post: I'm reading Vapnik's Statistical Learning Theory and have a question about the notation of an integral. He writes the loss functional for a classification problem with features [; x ;], labels [; omega in {0, dots, k } ;] , prediction function [; phi(x,alpha) ;] where [; alpha ;] is the vector of parameters to be learned by minimizing the functional, and with [; x ;] and [; omega ;] distributed according to pdf [; F( omega, x ) ;] as [; R(alpha) = int L( omega , phi(x, alpha) ) d F ( omega, x) ;] This is equation 1.6 on pg 25. My question is, what exactly does [; d F ( omega, x) ;] mean? And does this somehow allow us to integrate over [; omega ;] despite it being a discrete variable? To clarify the level of understanding of notation I'm coming from, I think I would have written this loss functional as [; R(alpha) = int sum_omega L(omega, phi(x,alpha)) F(omega, x) dx ;] |
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I suggest using MathURL website. There is a lengthy discussion on the or-exchange about this problem it may help you as well.
How do you write the markdown to make the MathURL link render in your metaoptimize comment?
(Sep 10 '12 at 15:58)
Ian Goodfellow
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dF(omega, x) is standard notation in measure theory. It means you're not integrating over the lebesgue measure for x but instead integrating over the measure F(omega, x). Read either this paper or the wikipedia page for lebesgue integration to get a better sense for what it means, but your intuition is correct for some non-pathological cases. About your main question, however, there's currently no acceptable way to write math on this site. If you think of an approach that should work, email Joseph Turian and he'll probably implement it.
(Jul 26 '10 at 15:11)
Alexandre Passos ♦
does the way I wrote it work for cases where the expression is Riemann integrable? ie, are these pathological cases the cases where the expression is Lebesgue integrable but not Riemann integrable?
(Jul 26 '10 at 17:08)
Ian Goodfellow
Exactly. Lebesgue-integrable is more general than Riemann-integrable, so if the function and the measure are Riemann-integrable you have nothing to worry about.
(Jul 26 '10 at 17:15)
Alexandre Passos ♦
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Man, I wish we could write math easily. I made a suggestion that MathJax get integrated, which seems to work pretty well with Markdown (I actually use it in my Markdown based blog to great success).