Editing Open Problems:81
Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.
The edit can be undone.
Please check the comparison below to verify that this is what you want to do, and then save the changes below to finish undoing the edit.
| Latest revision | Your text | ||
| Line 8: | Line 8: | ||
$$ | $$ | ||
for $\alpha\neq 1$, and $H_1(p) = \lim_{\alpha\to 1} H_\alpha(p)$. | for $\alpha\neq 1$, and $H_1(p) = \lim_{\alpha\to 1} H_\alpha(p)$. | ||
| − | In particular, $H_1$ corresponds to the Shannon entropy $H(p)=-\sum_{x\in\Omega} p(x) \log p(x)$. The problem of estimation Rényi entropy of a distribution $p$, given i.i.d. samples from it, was studied in {{cite|AcharyaOST-17}}, where the authors obtain tight bounds for every integer $\alpha\neq 1$, and nearly-tight ones for every non-integer $\alpha \geq 0$. In the later | + | In particular, $H_1$ corresponds to the Shannon entropy $H(p)=-\sum_{x\in\Omega} p(x) \log p(x)$. The problem of estimation Rényi entropy of a distribution $p$, given i.i.d. samples from it, was studied in {{cite|AcharyaOST-17}}, where the authors obtain tight bounds for every integer $\alpha\neq 1$, and nearly-tight ones for every non-integer $\alpha \geq 0$. In the later, however, the sample complexity is only known up to a subpolynomial factor: resolving the exact dependence on $\alpha$ and the alphabet size would be interesting. |
