Instructions: In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.
Input: Consider the following fundamental estimation problem: there are n entities, each with an unknown parameter pi [0, 1], and we observe n independent random variables,X1, . . . , Xn, withXi Binomial(t, pi). How accurately can one recover the "histogram" (i.e. cumulative density function) of the pis? While the empirical estimates would recover the histogram to earth mover distance ( 1 t ) (equivalently, '1 distance between the CDFs), we show that, provided n is sufficiently large, we can achieve error O( 1t ) which is information theoretically optimal. We also extend our results to the multi-dimensional parameter case, capturing settings where each member of the population has multiple associated parameters. Beyond the theoretical results, we demonstrate that the recovery algorithm performs well in practice on a variety of datasets, providing illuminating insights into several domains, including politics, and sports analytics.
Output:
Optimally Learning Populations of Parameters