Timeline for Approximating a probability distribution by a mixture
Current License: CC BY-SA 2.5
14 events
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| Jan 26, 2011 at 17:59 | comment | added | Jules | Can you give a little more information about the physical background of the problem? How can you mix even one $\int p(x)f(x)$? You seem to be mixing uncountably many things there? | |
| Jan 26, 2011 at 17:39 | comment | added | Jules | Perhaps Frechet derivatives on Banach spaces can help here. What you'd want to say is d{the L1 norm}/dp = 0, except that p is a function not a number. Frechet derivatives generalize taking derivatives to taking derivatives by functions. That is, if you have an operator H : (R -> R) -> R, you can find H'. The solution in this case would be the solution to H'(p) = 0. I believe that H'(p) = 0 will reduce to a differential equation for p. | |
| Sep 16, 2010 at 14:39 | comment | added | Zsbán Ambrus | The iteration algorithm called Expectation-maximization is often suitable for approximations with mixitures, though it might not actually converge to the minimum you asked for. | |
| Sep 16, 2010 at 4:15 | answer | added | ronaf | timeline score: 2 | |
| Sep 2, 2010 at 16:05 | comment | added | Anthony Leverrier | Unfortunately, it is really the $L^1$ distance which is relevant in my problem so I cannot switch from the $L^1$ to the $L^2$ distance. Furthermore, as the distributions are defined over $\mathbb{N}$, I cannot see how a bound on the $L^2$ distance could give any information concerning the $L^1$ distance? | |
| Sep 1, 2010 at 20:58 | comment | added | John Jiang | Maybe it's better to bound $L^1$ distance by $L^2$, then Fourier analytic techniques can be used. That is also the strategy used in length minimization via energy minimization, well known to differential geometers. | |
| Sep 1, 2010 at 20:49 | history | edited | Anthony Leverrier | CC BY-SA 2.5 |
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| Sep 1, 2010 at 16:47 | history | edited | Anthony Leverrier | CC BY-SA 2.5 |
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| Sep 1, 2010 at 14:03 | history | edited | Anthony Leverrier | CC BY-SA 2.5 |
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| Sep 1, 2010 at 12:08 | comment | added | Anthony Leverrier | I edited my question according to your remarks. | |
| Sep 1, 2010 at 12:07 | history | edited | Anthony Leverrier | CC BY-SA 2.5 |
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| Sep 1, 2010 at 11:02 | comment | added | robin girard | Can you motivate a bit more your problem (why do you need that in two lines, is it some sort of least favorable prior for simultaneous testing)? it is probability over $\mathbb{R}$ ? The norm you use in your sum is the $L^1$ norm between distribution right ? note that $p$ should have integral = 1. | |
| Sep 1, 2010 at 10:37 | history | edited | Anthony Leverrier | CC BY-SA 2.5 |
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| Sep 1, 2010 at 10:25 | history | asked | Anthony Leverrier | CC BY-SA 2.5 |