D\big)
& = & (1+\gamma)^{m_{1i}+m_{2i}+1}\dfrac{\lambda_i^{m_{1i}+m_{2i}}
\big[1-(1+\gamma)\lambda_i\big]^{m_{0i}}}{{\mbox{{\bf B}}}(m_{1i}+m_{2i}+1,m_{0i}+1)},
\quad0<\lambda_i<\dfrac{1}{1+\gamma},\qquad\end{aligned}$$ where ${\mbox{{\bf B}}}(.,.)$ refers to the Beta function. In other words, $\dfrac{1-\gamma}{1+\gamma}\sim{\mbox{{\bf Be}}}(m_{2+}+1,m_{1+}+1)$ and $(1+\gamma)\lambda_i|\gamma\sim{\mbox{{\bf Be}}}(m_{1i}+m_{2i}+1,m_{0i}+1),\,i=0,1$, where the notation ${\mbox{{\bf Be}}}(\alpha,\beta)$ represents the standard Beta distribution with shape parameters $\alpha$ and $\beta$. The uniform prior can be viewed as a process of adding $1/2$ to the summary statistics in the bottom four cells of the $3\times2$ table and 1 to the top two cells. The uniform prior is appealing in situations where the physical system imposes a natural parameterization with a nice physical interpretation. In general, the uniform distribution lacks the property of parameterization invariance.

Jeffreys’ Prior {#sub2sec4}
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Jeffreys’ prior has the property of being invariant under a one-to-one reparameterization [@Jeffreys1946]. Regardless of the parameterization used, Jeffreys’ prior distribution is proportional to the square root of the absolute value of the determinant of the Fisher’s information matrix.

Define $U = (1+\gamma)\lambda_0$ or equivalently $\lambda_0 = \dfrac
{U}{1+\gamma}$, and let $V = (1+\gamma)\lambda_1$ or equivalently $\lambda_1 =\dfrac
{V}{1+\gamma}$. Under this new parametrization, the parameter space reduces to the interval $(0,1)$ for each of the three parameters $\gamma$, $U$ and $V$. Thus, we are no longer dealing with a constrained parameter space. Moreover, 