We have n binomial distributions {$b_i$} - each with m trials, and a probability of success $p_i$ somewhere in the range [0,1]. Also, each binomial distribution $b_i$ is assigned some weight $w_i$, such that $\sum (w_i) = 1$. Let the distribution D be as follows - with probability $w_i$ we sample from distribution $b_i$.

In general, I will be interested to know how many samples I need to estimate all $w_i$ and $p_i$.

Also assuming $\exists \bar{i}$ s.t. $w_\bar{i} > \frac{1}{2^n}$ and $\forall i \neq \bar{i}$ $p_\bar{i} > p_i +\frac{2}{n}$, Is there a way, with fewer samples, to estimate $p_\tilde{i}$?


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