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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 $$p(\alpha)=\lambda_\alpha\exp(-\lambda_\alpha \alpha)$$$$p(\beta)=\lambda_\beta\exp(-\lambda_\beta \beta)$$
But this is just an example. Any non-negative prior is an option. Modern Bayesian inference software such as Stan dose not restrict you to conjugate priors.
