Timeline for Maximizing the expectation of a polynomial function of iid random variables
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 8 at 8:32 | comment | added | Tobias Fritz | @Dan: I did not know this question and recently asked a very similar one. I developed essentially the same method as you did here and applied it to my case. Has you or anyone else written this up? I couldn't find any literature about it, which seemed surprising, and ended up writing a draft myself (with a proof that is simpler and more general). And now this question was pointed out to me! Shoot me an email in case that you're still interested in this. | |
| Feb 26, 2019 at 13:18 | comment | added | Dan | @student This is a good way to see it. Indeed, the function I called $\widehat{f}_k$ is the integral of $f$ with respect to $\frac{1}{k!}\sum_{\sigma \in S_k}\delta_{(x_{\sigma(1)},\ldots,x_{\sigma(n)})}$. In fact, this discrete measure is approximately equal to the product measure $$\frac{1}{k^n}\sum_{i_1,\ldots,i_n=1}^k \delta_{(x_{i_1},\ldots,x_{i_n})} = \left( \frac{1}{k}\sum_{i=1}^k \delta_{x_i}\right)^n$$ in the sense that the total variation distance between the two measures converges to zero as $k\to\infty$ with $n$ fixed, uniformly in the choices of $x_i$'s. | |
| Feb 25, 2019 at 17:54 | comment | added | Paata Ivanishvili | If I did not make a mistake in my calculation then the right condition in case of n=3 for symmetric $f(x,y,z)$ is that $\int f(x,y,z) d\mu(x) d\mu(y)$ is independent of $z$. Can it help to solve the "baby" problem? | |
| Feb 25, 2019 at 17:24 | comment | added | Romeo | @PaataIvanishvili Nice observation about the semi-circle law! I now see the analogy with S. Lee's answer. I did want to try a similar approach but I thought that since we do not have any longer a "scalar product" that was not meaningful. Probably the correct way of writing it is by observing that some integrals has to be independent of the other variables... I'm going to try in a moment and I'll update if I have some news. Thanks! | |
| Feb 25, 2019 at 16:24 | comment | added | Paata Ivanishvili | @Romeo, for example, following Sangchul Lee if we are given symmetric $f(x,y,z)$, and assume the maximizer $d\mu(x)$ exists, then consider a perturbed maximizer $\mu \mapsto \mu+\varepsilon \delta_{s} - \varepsilon \delta_{t}$ and differentiate in $\varepsilon$ and make it zero. What condition you get on $d\mu$? | |
| Feb 25, 2019 at 16:24 | comment | added | Paata Ivanishvili | @Romeo notice that for this symmetric function $f(x,y)=\ln|x-y|-x^{2}-y^{2}$ and the semicircle measure $d\mu(x)$ we have $\int_{-1}^{1}f(x,y)d\mu(x)$ is independent of $y$. This correct "property" was explained by Sangchul Lee in his series of answers to previous problems. It seems to me that in general $n>2$ something similar should hold. | |
| Feb 25, 2019 at 14:53 | comment | added | user114668 | I am trying to understand this result. We can write $f(x_{\sigma(1)} \ldots x_{\sigma(n)})$ as an integral over $f$ and delta distributions at $x_{\sigma(1)} \ldots x_{\sigma(n)}$, $$\int \! f(x_1' \ldots x_n') \, \Bigl\{\frac{1}{k!}\sum_{\sigma \in S_k} \delta(x_1' - x_{\sigma(1)}) \cdots \delta(x_n' - x_{\sigma(n)}) \Bigr\} \, d^nx' \;.$$ So has this essentially shown that there is a maximising distribution, and that it can be approximated by sufficiently many point masses? | |
| Feb 25, 2019 at 9:02 | comment | added | Romeo | @Dan Thanks for your comments. I appreciate your contribution and I do not find it useless. Indeed, I decided to award the bounty to you, as you provided me a lot of new tools and theorem to think about. I hope the others who provided an answer do not mind: I acknowledge them too, of course, but your answer is more in the spirit of what I was looking for. | |
| Feb 25, 2019 at 8:58 | history | bounty ended | Romeo | ||
| Feb 24, 2019 at 16:12 | comment | added | Dan | @Romeo The more I reflect on this answer the more useless it seems. Very different things can happen in different examples, so you should not expect a very general answer. An important example from random matrix theory involves the "non-commutative entropy": Take $n=2$ and $f(x,y) = \log|x-y| - x^2 - y^2$. Then, over all probability measures $\mu$ on $[-1,1]$, the value of $\int_{[-1,1]^2}f(x,y)\mu(dx)\mu(dy)$ is maximized by the semicircle law $\mu(dx) = \frac{2}{\pi}\sqrt{1-x^2}dx$. See e.g. Theorem 1.3 here: link.springer.com/article/10.1007/s004400050119 | |
| Feb 24, 2019 at 16:10 | comment | added | Dan | @Sinusx The other way around is immediate because $C_k \subset P_e(E^k)$. | |
| Feb 22, 2019 at 23:38 | comment | added | Viktor B | '...says that for each $\eta \in P_e(E^k)$ there exists $\nu \in C_k$ ...' For your argument to work, shouldn't it be the other way around: for each $\nu \in C_k$ there exists $\eta \in P_e(E^k)$, and so on? | |
| Feb 22, 2019 at 16:38 | comment | added | Romeo | Gosh, thanks a lot for such an answer! I am very happy to read your contribution. I am not an expert in probability (I am not familiar with De Finetti or Diaconis-Friedman theorem) so I need sometime to study and understand your approach, which looks interesting. As a "concrete" example, may I ask you to kindly give a look here (where actually the whole story began)? Probably in that specific example your approach can yield the final answer... Thanks again! | |
| Feb 22, 2019 at 16:22 | history | edited | Dan | CC BY-SA 4.0 |
deleted 4 characters in body
|
| Feb 22, 2019 at 14:51 | history | answered | Dan | CC BY-SA 4.0 |