The likelihood function is a Binomial one -- say $k$ failures in $n$ trails given the probability of failure is $p$.
The $p$ is a function of two independent r.v. $x$ and $y$: $p=a-bx+cy$ where $a$, $b$ and $c$ are known constants in the range of $[0,1]$, $[0, \infty)$ and $[0, \infty)$ respectively. $x$ and $y$ are both in the range of $[0,1]$.
Now the problem is, once I observe some data $k$ anf $n$, I want to do bayesian inference on the $x$ and $y$. I am thinking to use conjugage but cannot find good priors for them... even good approximation is ok.
Also I see there is the problem that the range of $p$ could be out of $[0,1]$, is there anyway we somehow choose the priors that can change/rescale the range of $x$ and $y$ to make sure $p \in [0,1]$?