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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 $p \in [0,1]$?

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I am wondering how you can make sure that $0\leq p\leq 1$ with the given conditions. We can easily see that $a-bx+cy$ can range from $-\infty$ to $+\infty$.

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  • $\begingroup$ you are quite right, I will re-edit the question $\endgroup$ – weidade3721 Nov 28 '19 at 20:57

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