and $1/2$ to the top two cells.

@GhoshMukerjee1992 advise reversing the role of parameters of interest and nuisance parameters to obtain a reverse reference prior. That is, reconsider the group ordering of $\{\gamma\}$ and then $\{U,V\}$ or $\{\gamma\}$ and then $\{V,U\}$. The reference prior remains unchanged as shown in Appendix 2. We also discuss the idea of reference priors under the partial information introduced by @SunBerger1998 and known as conditional reference priors in Appendix 2.

Sampling from the Posterior Distribution {#sub4sec4}
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With analytical solutions difficult to derive or compute, we turn to Monte-Carlo simulation methods. We first discuss how to simulate $\gamma$ from the distribution $$\begin{aligned}
f(\gamma) & = &
\dfrac{2^{\mu}}{{\mbox{{\bf B}}}(\mu,\nu)}\dfrac{\gamma^{\mu-1} \;(1-\gamma)^{\nu-1}}{
(1+\gamma)^{\mu+\nu}}\,,\qquad0<\gamma<1,\end{aligned}$$ which includes the marginal posterior distributions, $\pi_U(\gamma\,|\,
D)=\pi_J(\gamma\,|\,D)$, as special cases with $\mu=m_{1+}+\tfrac{1}{2}$ and $\nu=m_{2+}+\tfrac{1}{2}$. Then, we discuss how to simulate $(\lambda_1,\lambda_0)$ jointly from $f(\lambda_1,\lambda_0 \mid D)$.

We propose two direct and efficient approaches to generate $\gamma$ from $f(\gamma)$.

-   Let $\gamma= \dfrac{e^\Psi}{1+e^\Psi}$. We show in Appendix 3 that $\phi= \Psi-\log(2)$ has the same distribution as $\mbox{logit}(p) =
    \log\left(
    \dfrac{p}{1-p}\right)$, where $p\sim{\mbox{{\bf Be}}}(\mu,\nu)$. Thus, simulate $p_i\sim{\mbox{{\bf Be}}}(\mu,\nu)$, compute $\phi_i=\mbox{logit}(p_i)-\log(2)$, and set $\gamma_i=\dfrac{e^{\phi_i}}{1+e^{\phi_i}},\,i=1,\dots,M$.

-   Let $\gamma= \dfrac{1-\pi}{1+\pi}$. We show in Appendix 3 that $\pi\sim{\mbox{{\bf Be}}}(\nu,\mu)$. Thus, 