context
In Factor graphs on discrete variables, the parameters are contained in factors associated each with a subset of the random variables in the system. Each factor provides a different positive parameter for each valuation of the subset.
If you are familiar with Bayesian or Markov networks, you probably know that it is usual (when learning models) to use priors over those parameters and to consider the parameters being mutually independent and sharing the same distribution $d$. The usual distributions are associated with regularizations (L1-regularisation supposes the log-parameters have a Laplace distribution, while L2-regularisation supposes the log-parameters have a normal distribution).
I am curious to see the link between those priors and the Dirichlet priors on a categorical/discrete variable. In the general case, a factor graph enforces independencies between variable so the overall probability cannot have a Dirichlet prior. But in the very simple case when there exists a single factor over all variables, the two approaches can be compared. I have started with the simpliest case possible : a single boolean random variable.
problem
In that case, there are only two parameters : $R_F$ and $R_T$ that can be seen as unormalized probabilities so the probability of the variable to be True is $p = \frac{R_T}{R_T+R_F}$. (with $R_T \gt 0 $ and $R_F \gt 0 $)
My aim is to find a distribution $d$ for my two parameters so that the resulting distribution for $p$ is a beta distribution with two identical parameters ($p \sim Beta(\alpha,\alpha)$). The case where $\alpha = 1$ i.e. p is uniformly distributed is maybe simplier and can be a useful first step.
What I have tried
The difficulty comes from the redundancy of the parameters. It is feasible to deduce the distribution of $p$ from that of $R_T$ and $R_F$. But I want to do the opposite.
If I remove the redundancy by setting $R_F=1$, then $R_T$ is the odd related to $p$ and the beta prime distribution is what I am looking for. I have tried this distribution in my original problem, but without success.
