51ÂÜÀò

Consider the problem of estimating the Shannon  entropy of a distribution on $ °ì$Ìý elements from $ ²Ô$Ìý ¾±²Ô»å±ð±è±ð²Ô»å±ð²Ô³ÙÌý samples. We show that the minimax mean-square error is within  universal multiplicative constant factors of  $\left( \frac{n}{k \log n} \right)^{2} + \frac{\log^2 k}{n}$. °Õ³ó¾±²õÌý implies the recent result of Valiant-Valiant [ 1 ] that the minimal  sample size for consistent entropy estimation scales according to $\Theta( \frac{k}{\log k} )$.  The apparatus of best polynomial approximation plays a key role in both the minimax lower bound and the construction of optimal estimators.