 Solution

Take flat priors on $\alpha ,\beta ,\mu$ and scale priors $\pi ({\sigma }^2,{\tau }^2)\propto ({\sigma }^2{\tau }^2{)}^{-1}$ on the positive variances. The posterior distribution is thenA random-walk Metropolis–Hastings algorithm can update $(\alpha ,\beta ,\mu ,\log {\sigma }^2,\log {\tau }^2)$ with a symmetric proposal and accept a proposed state $z^{\prime }$ from $z$ with probability $1\wedge \pi (z^{\prime }\mid \mathcal{D})/\pi (z\mid \mathcal{D})$, including the Jacobian if the target is represented in transformed coordinates. Its transition kernel satisfieswhich is detailed balance; hence the posterior is stationary.