& \; \pi_{H_1}\big(\gamma, u, v\big) \\
= & \; \dfrac{1}{K} \dfrac{2^{\frac{1}{2}}}{{\mbox{{\bf B}}}(\frac{1}{2},\frac{1}{2})}
\dfrac{\gamma^{\frac{1}{2}-1} (1-\gamma)^{\frac{1}{2}-1}}{(1+\gamma)} (u+rv)^{d}
\dfrac{u^{\frac{1}{2}-1}(1-u)^{\frac{1}{2}-1}}{{\mbox{{\bf B}}}(\frac{1}{2}, \frac
{1}{2})} \dfrac{v^{\frac{1}{2}-1}
(1-v)^{\frac{1}{2}-1}}{{\mbox{{\bf B}}}(\frac{1}{2}, \frac{1}{2})},\end{aligned}$$ where $K$ is the normalizing constant and $0<\gamma, u, v<1$. Note that when $d=0$, $K=1$. Two choices of $d$ are of interest: $d=0$ corresponding to the reference prior and $d=1/2$ corresponding to Jeffreys’ prior. The marginal predictive distribution under $H_1$ is $$\begin{aligned}
& \; p_{H_1}(m_{10}, m_{20}, m_{11}, m_{21}) \\
= & \; \dfrac{1}{K} p(m_{10}, m_{20}, m_{11}, m_{21})
\int_0^1 \int_0^1 (u+rv)^{d}
\dfrac{u^{m_{10}+m_{20}+\frac{1}{2}-1} (1-u)^{m_{00}+\frac{1}{2}-1}}{{\mbox{{\bf B}}}\big(m_{10}+m_{20}+\frac{1}{2},m_{00}+\frac{1}{2}\big)} \\
& \; \times
\dfrac{v^{m_{11}+m_{21}+\frac{1}{2}-1} (1-v)^{m_{01}+\frac{1}{2}-1}}{{\mbox{{\bf B}}}\big(m_{11}+m_{21}+\frac{1}{2},m_{01}+\frac{1}{2}\big)} dv\,du,\end{aligned}$$ where $$\begin{aligned}
p(m_{10}, m_{20}, & m_{11}, m_{21}) = {m_{+0} \choose{m_{00}, m_{10}, m_{20}}}
{m_{+1} \choose{m_{01}, m_{11}, m_{21}}} \dfrac{{\mbox{{\bf B}}}\big(m_{1+}+\frac
{1}{2}, m_{2+}+\frac{1}{2}\big)}{{\mbox{{\bf B}}}\big(\frac{1}{2},\frac{1}{2}\big)} \\
& \; \times\dfrac{{\mbox{{\bf B}}}\big(m_{10}+m_{20}+\frac{1}{2}, m_{00}+\frac
{1}{2}\big)}{{\mbox{{\bf B}}}\big(\frac{1}{2},\frac{1}{2}\big)}
\dfrac{{\mbox{{\bf B}}}\big(m_{11}+m_{21}+\frac{1}{2}, m_{01}+\frac{1}{2}\big)}{{\mbox{{\bf B}}}\big
(\frac{1}{2},\frac{1}{2}\big)}.\end{aligned}$$ Under $H_0$, the likelihood reduces to $$L(\gamma, \theta) = \left\{\prod_{i=0}^1 {m_{+i} \choose{m_{0i},
m_{1i}, m_{2i}}}\right\} \;2^{m_{1+}} \dfrac{\gamma^{m_{1+}}
(1-\gamma)^{m_{2+}}}{(1+\gamma)^{m_{1+}+m_{2+}}}\,
\theta^{m_{1+}+m_{2+}}(1-\theta)^{m_{0+}},$$ where $\theta=(1+\gamma)\lambda$. Jeffreys’ and Bernardo’s priors belong to the family of priors $$\begin{aligned}
\pi_{H_0}(\gamma, \theta) & = & \dfrac{2^{\frac{1}{2}}}{{\mbox{{\bf B}}}(\frac
{1}{2},\frac{1}{2})}\,
\dfrac{\gamma^{\frac{1}{2}-1}\;(1-\gamma)^{\frac{1}{2}-1}}{(1+\gamma)}
\;
\dfrac{\theta^{a-1}\;(1-\theta)^{\frac{1}{2}-1}}{{\mbox{{\bf B}}}(a,\frac
{1}{2})},\quad0<\gamma,\theta<1.\end{aligned}$$ Thus, the resulting marginal predictive distribution is $$\begin{aligned}
& \; p_{H_0}(m_{10}, m_{20}, m_{11}, m_{21}) \\
= & \; {m_{+0} \choose{m_{00},
m_{10}, m_{20}}}\;{m_{+1} \choose{m_{01},m_{11}, m_{21}}}\,
\dfrac{{\mbox{{\bf B}}}\big(m_{1+}+\frac{1}{2},\, m_{2+}+\frac{1}{2}\big)}{{\mbox{{\bf B}}}\big
