orderings: (i) $\{U,V\}$ and then $\{\gamma
\}$ and (ii) $\{V,U\}$ and then $\{\gamma\}$.

\[prop1sub3sec4\] Bernardo’s reference prior corresponding to the groups ordering $\{U,V\}
$ and then $\{\gamma\}$ or $\{V,U\}$ and then $\{\gamma\}$ is $$\label{eq1sub3sec4}
\pi_R(\gamma, u, v) = \dfrac{2^{1/2}\gamma^{1/2-1}
(1-\gamma)^{1/2-1}}{{\mbox{{\bf B}}}(1/2,1/2)(1+\gamma)}
\dfrac{u^{1/2-1}(1-u)^{1/2-1}}{{\mbox{{\bf B}}}(1/2,1/2)} \dfrac
{v^{1/2-1}(1-v)^{1/2-1}}{{\mbox{{\bf B}}}(1/2,1/2)},\quad$$ $0<\gamma, u, v<1$. That is, $\dfrac{1-\gamma}{1+\gamma}\sim{\mbox{{\bf Be}}}(1/2,1/2)$, $U\sim{\mbox{{\bf Be}}}(1/2,1/2)$ and $V\sim\break{\mbox{{\bf Be}}}(1/2,1/2)$.

\[prop2sub3sec4\] In the original parameterization, $(\gamma, \lambda_0, \lambda_1)$, Bernardo’s reference prior is equivalent to $$\begin{aligned}
\label{eq2sub3sec4}
\pi_R(\gamma, \lambda_0, \lambda_1) = & \;
\dfrac{\sqrt{2}}{\pi} \gamma^{1/2-1}
(1-\gamma)^{1/2-1} \dfrac{\lambda_0^{1/2-1}
\big[1-(1+\gamma)\lambda_0\big]^{1/2-1}}{{\mbox{{\bf B}}}(1/2,1/2)}\nonumber\\
& \; \times\dfrac{\lambda_1^{1/2-1}
\big[1-(1+\gamma)\lambda_1\big]^{1/2-1}}{{\mbox{{\bf B}}}(1/2,1/2)},\qquad
(\gamma, \lambda_0, \lambda_1) \ \in\Omega.\end{aligned}$$

The proofs of Propositions \[prop1sub3sec4\] and \[prop2sub3sec4\] are given in Appendix 2. Unlike Jeffreys’ prior, $\gamma$, $U$, and $V$ are independent under Bernardo’s reference prior. The posterior distribution resulting from the use of Bernardo’s reference prior is $$\begin{aligned}
& \; \pi_R\big(\gamma, \lambda_0, \lambda_1|D\big) \\
\propto& \;
\dfrac{2^{m_{1+}+1/2}}{{\mbox{{\bf B}}}(m_{1+}+1/2,m_{2+}+1/2)}\gamma^{m_{1+}+1/2-1}
(1-\gamma)^{m_{2+}+1/2-1}\lambda_0^{m_{10}+m_{20}+1/2-1}\\
& \; \times\big[1-(1+\gamma)\lambda_0\big]^{m_{00}+1/2-1}
\lambda_1^{m_{11}+m_{21}+1/2-1}\big[1-(1+\gamma)\lambda_1\big]^{m_{01}+1/2-1},\end{aligned}$$ $(\gamma,\lambda_0,\lambda_1)\in\Omega$ or equivalently $$\begin{aligned}
& \; \pi_R\big(\gamma, u, v \,|D) \\
= & \; \dfrac{2^{m_{1+}+1/2}}{{\mbox{{\bf B}}}(m_{1+}+1/2, m_{2+}+1/2)}
\dfrac{\gamma^{m_{1+}+1/2-1} (1-\gamma)^{m_{2+}+1/2-1}}{(1+\gamma
)^{m_{1+}+m_{2+}+1}} \nonumber\\
& \; \times\dfrac{u^{m_{10}+m_{20}+1/2-1}(1-u)^{m_{00}+1/2-1}}{{\mbox{{\bf B}}}(m_{10}+m_{20}+1/2, m_{00}+1/2)}
\dfrac{v^{m_{11}+m_{21}+1/2-1} (1-v)^{m_{01}+1/2-1}}{{\mbox{{\bf B}}}(m_{11}+m_{21}+1/2, m_{01}+1/2)},\end{aligned}$$ $0<\gamma,u,v<1$. The reference prior can be viewed as adding $1/4$ to each of the bottom four cells of the $3\times2$ table 