I have this particular equation
$\frac{(\alpha-1)!(\beta-1)!}{(\alpha+\beta-1)!} = \frac{\Gamma(p)(1+q)^{n+2p} 2^n}{q^{p}(2+q)^{n+p}}$.
Now, given the values of $\alpha$ and $\beta$, I need to find the corresponding values of p and q. n is fixed for a given model. How do I find an analytical solution?
What I did so far.
Suppose, $A = \frac{(\alpha-1)!(\beta-1)!}{(\alpha+\beta-1)!}$, we can rewrite the above equation as,
$A q^{p}(2+q)^{n+p} = 2^n (p-1)!(1+q)^{n+2p}$.
After setting the values for A and n, I took LHS as a function and RHS as another function. Then I plotted the two functions and checked for intersections. While this method may work when the values are chosen properly, it's desirable to get an analytical solution.
Details:
$\alpha$ and $\beta$ are the parameters of the beta prior and it is known that $p >0, q >0$. n specifies the model ($n > 0$).
Now given the beta prior (i.e., the values of $\alpha$ and $\beta$), I need to find the values of p and q. How do I solve this?
Thanks in advance.