# Univariate priors for the parameters of a Beta distribution

I need a rather a prior on the parameters of a Beta distribution (i.e. $\alpha$ and $\beta$). I have an external constraint that requires me to use univariate priors, one for $\alpha$ and one $\beta$.

Ideally I would like to use two univariate priors that together are as close as possible to something like $p(\alpha,\beta)∝(\alpha+\beta)^{−5/2}$ (for anyone interested in this particular choice, see this thread)

What priors can I use for them?

Any prior on $$\alpha$$ (or $$\beta$$) is admissible as long as it satisfies the requirements of the beta distribution in your parameterization, usually $$\alpha >0$$ and $$\beta >0$$, and as long as it yields a finite posterior. Assuming univariate priors and independence of $$\alpha$$ and $$\beta$$, one option might be the exponential distribution, since it's bounded by $$0$$. Additionally, it has a mode at $$0$$, meaning that plausible values will tend to be small. Some might find this attractive because they may desire only vague prior information. In this case, your prior is $$p(\alpha)=\lambda_\alpha\exp(-\lambda_\alpha \alpha)$$$$p(\beta)=\lambda_\beta\exp(-\lambda_\beta \beta)$$