simulate $\pi_i\sim{\mbox{{\bf Be}}}(\nu, \mu)$ and set $\gamma_i = \dfrac{1-\pi_i}{1+\pi_i},\,i=1,\dots,M$.

We now focus on the joint marginal posterior distribution of $(U,V)$ obtained under Jeffreys’ prior $$\begin{aligned}
f(u, v\,|\,D) \propto& \; (u + rv)^{1/2}\,
\dfrac{u^{m_{10}+m_{20}-1/2}(1-u)^{m_{00}-1/2}}{{\mbox{{\bf B}}}(m_{10}+m_{20}+1/2,
m_{00}+1/2)}\,\\
& \times\dfrac{v^{m_{11} + m_{21}-1/2} (1-v)^{m_{01}-1/2}}{{\mbox{{\bf B}}}(m_{11}+m_{21}+1/2, m_{01}+1/2)},\end{aligned}$$ which is independent of $\gamma$. To simulate $M$ observations $(u_i,v_i), \; i=1,\cdots,M$, we proceed as follows:

-   Simulate independent observations $(u_i, v_i), \; i=1,\cdots
    ,M$, with $u_i\sim
    {\mbox{{\bf Be}}}\big(m_{10}+m_{20}+1/2, m_{00}+1/2\big)$ and $v_i\sim
    {\mbox{{\bf Be}}}\big(m_{11}+m_{21}+1/2, m_{01}+1/2\big)$.

-   Compute the weights $w_i=(u_i+rv_i)^{1/2}, \; i=1,\cdots,M$.

-   Use the acceptance/rejection sampling method: Simulate $\xi
    _i\sim U(0,1)$ and accept the pair $(u_i,v_i)$ only if $\xi_i < w_i/(1+r)$.

-   Or use the importance sampling method, where all the pairs $(u_i,v_i)$ are accepted and use the weights $w_i$ to correct for the bias in the computation of posterior mean and quantiles.

Under the reference prior, we simulate independent observations $(u_i,
v_i), \; i=1,\cdots,M$, with $u_i\sim
{\mbox{{\bf Be}}}\big(m_{10}+m_{20}+1/2, m_{00}+1/2\big)$ and $v_i\sim
{\mbox{{\bf Be}}}\big(m_{11}+m_{21}+1/2, m_{01}+1/2\big)$. Having simulated a triplet $(\gamma_i, U_i,V_i)$, we compute $\lambda^i_0=U_i/(1+\gamma_i)$ and $\lambda^i_1=V_i/(1+\gamma_i)$ as well as the risk difference, $\Delta
_i=\dfrac{(V_i-U_i)}{(1+\gamma_i)}$, the risk ratio, $R_i=\dfrac{V_i}{U_i}$ (does not depend on $\gamma$), and the odds ratio, $\psi_i=\dfrac{V_i\big[1+\gamma_i-U_i\big]}{U_i\big
[1+\gamma_i-V_i\big]}$. These simulated values are in turn used to 