-1
$\begingroup$

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.

$\endgroup$
2
  • 1
    $\begingroup$ 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
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.