Suppose I have $i$ coins, all of which are weighted to have a different probability $p$ of flipping heads. This results in $i$ Bernoulli distributions with different $p_i$. Cumulatively, this results in a Poisson Binomial distribution. For this scenario, $i$ is very large, so normal approximation appears to perform well (as well as other factors). However, I want to weigh each Bernoulli distribution equal to the amount evidence that supports it ($w_i$), which has been predetermined prior to this coin-flipping experiment. The frequency of $w_i$ are exponentially distributed in this case. How would I go about hypothesis testing? If I multiply $p_i * w_i$, it results in a loss of my Gaussian distribution. So how should I go about this? I found this paper, but it is slightly above my comprehension level: http://www.mscs.mu.edu/~jsta/issues/11(4)/JSTA11(4)p3.pdf Advice and feedback would be greatly appreciated!
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