Suppose a friend has calculated a posterior distribution from a Beta prior and binomial likelihood, and you are interested in the prior parameters they used, but they won't give them to you. They only provide the Beta parameters ($\alpha, \beta$) for a posterior distribution they have calculated and the $p$ from the binomial likelihood they used to calculate it, but no $n$ or $k$.

Is it possible to calculate the Beta prior parameters they used to generate the posterior?

My intuition is that you could do something like:

$$\alpha - p\alpha$$ $$\beta - (1-p)\beta$$

My thinking is that this would be akin to subtracting the evidence from the posterior, on the same scale as the posterior, but I don't have a good proof for this.

Any thoughts would be appreciated.

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    $\begingroup$ When you say $p$ value do you mean some sort of $p$-value, or the proportion $p=k/n$ or something else. $\endgroup$ – Thomas Lumley Sep 14 '20 at 4:09
  • $\begingroup$ Sorry, using the word "value" made it ambiguous. I mean the proportion $k/n$. I have removed the word "value". $\endgroup$ – user3242357 Sep 14 '20 at 4:14

It's not possible.

Let $\alpha_0$ and $\beta_0$ be the prior parameters. We have $\alpha=\alpha_0+k$ and $\beta=\beta_0+(n-k)$. If you had $p=1/2$, $\alpha=10$, $\beta=10$, you could have had $k=1$, $n=2$, $\alpha_0=\beta_0=9$ or $k=9$, $n=18$, $\alpha_0=\beta_0=1$, or lots of other less symmetric possibilities (eg $k=5$, $n=14$, $\alpha_0=5$, $\beta_0=1$).

If you knew some other piece of information like $\alpha_0+\beta_0$ you could do it.


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