# Reverse engineering Beta prior parameters from Binomial likelihood and posterior beta parameters

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.

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

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.