priors under consideration is $$\begin{aligned}
\pi_{H^\ast_1}(\gamma_0, \gamma_1, u, v) = &
\dfrac{2^{\frac{1}{2}}\gamma_0^{\frac{1}{2}-1}\,
(1-\gamma_0)^{\frac{1}{2}-1}}{{\mbox{{\bf B}}}(\frac{1}{2},\frac{1}{2})(1+\gamma_0)}\;
\dfrac{2^{\frac{1}{2}}\gamma_1^{\frac{1}{2}-1}\,
(1-\gamma_1)^{\frac{1}{2}-1}}{{\mbox{{\bf B}}}(\frac{1}{2},\frac{1}{2})(1+\gamma_1)}\,
\dfrac{u^{a_0-1}\,(1-u)^{\frac{1}{2}-1}}{{\mbox{{\bf B}}}(a_0,\frac{1}{2})}\\
&\, \times\dfrac{v^{a_1-1}\,(1-v)^{\frac{1}{2}-1}}{{\mbox{{\bf B}}}(a_1,\frac
{1}{2})},\,\qquad0<\gamma_0, \gamma_1, u, v <1.\end{aligned}$$ The resulting marginal predictive distribution is $$\begin{aligned}
p_{H^\ast_1}\big(m_{10}, m_{20}, m_{11}, m_{21}\big) = & {m_{+0}
\choose{m_{00},
m_{10}, m_{20}}}\;{m_{+1} \choose{m_{01}, m_{11}, m_{21}}}\,\dfrac{{\mbox{{\bf B}}}\big(m_{10}+\frac{1}{2},
m_{20}+\frac{1}{2}\big)}{{\mbox{{\bf B}}}\big(\frac{1}{2},\frac{1}{2}\big)} \\
& \times\dfrac{{\mbox{{\bf B}}}\big(m_{11}+\frac{1}{2}, m_{21}+\frac{1}{2}\big)}{{\mbox{{\bf B}}}\big(\frac{1}{2},\frac{1}{2}\big)}\;
\dfrac{{\mbox{{\bf B}}}(m_{10}+m_{20}+a_0, m_{00}+\frac{1}{2})}{{\mbox{{\bf B}}}\big(a_0,\frac
{1}{2}\big)}\\
& \times\dfrac{{\mbox{{\bf B}}}(m_{11}+m_{21}+a_1, m_{01}+\frac{1}{2})}{{\mbox{{\bf B}}}\big
(a_1,\frac{1}{2}\big)}.\end{aligned}$$ We are only concerned with two sets of parameters choices: $d=1/2, a_0=a_1=1$ used in Jeffreys’ prior and $d=0, a_0=a_1=1/2$ used in the reference prior.

The Bayes factor for testing $H^\ast_0$ vs $H^\ast_1$ under the condition $P(H_1)=P(H_0)=\tfrac{1}{2}$ is $$\begin{aligned}
& BF_\gamma = \dfrac{p_{H_1}\big(m_{10}, m_{20}, m_{11}, m_{21}\big
)}{p_{H^\ast_1}
\big(m_{10}, m_{20}, m_{11}, m_{21}\big)} \\
= & \; \dfrac{{\mbox{{\bf B}}}\big(m_{1+}+\frac{1}{2}, m_{2+}+\frac{1}{2}\big) {\mbox{{\bf B}}}\big
(a_0,\frac{1}{2}\big)\,{\mbox{{\bf B}}}\big(a_1,\frac{1}{2}\big)}
{{\mbox{{\bf B}}}\big(m_{10}+\frac{1}{2}, m_{20}+\frac{1}{2}\big) {\mbox{{\bf B}}}\big(m_{11}+\frac
{1}{2}, m_{21}+\frac{1}{2}\big)
{\mbox{{\bf B}}}\big(\frac{1}{2},\frac{1}{2}\big)}\\
& \times\dfrac{{\mbox{{\bf B}}}\big(m_{10}+m_{20}+\frac{1}{2}, m_{00}+\frac{1}{2}\big)
{\mbox{{\bf B}}}\big(m_{11}+m_{21}+\frac{1}{2}, m_{01}+\frac{1}{2}\big)}
{{\mbox{{\bf B}}}\big(m_{10}+m_{20}+a_0, m_{00}+\frac{1}{2}\big)
{\mbox{{\bf B}}}\big(m_{11}+m_{21}+a_1, m_{01}+\frac{1}{2}\big)}\\
& \times\int_0^1 \int_0^1 \Big\{ \dfrac{(u+rv)^{d}}{K} \dfrac
{u^{m_{10}+m_{20}+\frac{1}{2}-1}
(1-u)^{m_{00}+\frac{1}{2}-1}}{{\mbox{{\bf B}}}\big(m_{10}+m_{20}+\frac
{1}{2},m_{00}+\frac{1}{2}\big)}\\
& \hspace*{0.3in} \times\dfrac{v^{m_{11}+m_{21}+\frac{1}{2}-1}
(1-v)^{m_{01}+\frac{1}{2}-1}}{{\mbox{{\bf B}}}\big(m_{11}+m_{21}+\frac
{1}{2},m_{01}+\frac{1}{2}\big)} \Big\} dvdu.\end{aligned}$$ Under Jeffreys’ prior, $BF_\gamma$ is computed using computer simulation while under the reference prior it is computed exactly.

Comparisons of Bayesian and Frequentist Intervals: An Empirical Study {#sec5}
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In this section, 