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.