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I am having the following problem. I observe a random variable $Y \sim \text{Bern}(p)$, which is the product of two other independent random variables $X_1 \sim \text{Bern}(p_1)$ and $X_2 \sim \text{Bern}(p_2)$. I know that the original parameter is $p = p_1 \cdot p_2$. Is there any way to estimate $p_1$ and $p_2$?

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Assuming that you are only observing the variable $y$, the probabilities $p_1$ and $p_2$ are not identifiable in your model; only the parameter $p= p_1 \cdot p_2$ is identifiable. This means that you essentially have no information with which to properly estimate $p_1$ and $p_2$.

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