Binomial likelihood, the posterior with a conjugate beta prior is:
\begin{equation}
p(θ | y)=\frac{1}{B(\overline{α}, \overline{β})}
θ\overline{α-1}(1-θ)\overline{β-1}
\end{equation}
with
\begin{equation}
\begin{aligned} \overline{α} &=α0+y
\overline{β} &=β0+n-y \end{aligned}
\end{equation}
The beta prior can be specified as: \begin{equation} ≡ \text { binomial experiment with }\left(α0-1\right) \text { successes in }\left(α0+β0-2\right) \end{equation}
Check PPD for binomial likelihood on p. 151. We should take into
account sampling variability of
Contour probability: posterior evidence of $H0$ with HPD interval. Defined as: \begin{equation} P\left[p(θ | \boldsymbol{y})>p\left(θ0 | \boldsymbol{y}\right)\right] ≡\left(1-pB\right) \end{equation}
$pB$ is computed from the smallest HPD interval containing $θ0$.
$\operatorname{Beta}\left(α0, β0\right)$ prior is equivalent to a binomial experiment with $α0 - 1$ successes in ($α0 + β0 - 2$) experiments.
The non-informative beta prior has $α0=1, β0=1$ and is
equal the uniform prior on
Popular priors for BGLIM are normal proper priors with large variance. Gelman et al. however suggest Cauchy density with center 0 and scale parameter 2.5 for standardized continuous covariates.