There is a Bayesian linear model $Y$~$N\left(\beta_0\mathbf{1}+X\beta,\sigma^2I\right)$, and the setting for spike-and-slab prior is as follows:
$\left.\beta_i\ \right|\ \gamma_i,\tau_i,\phi_i\ $~$\ \left(\gamma_i\right)\bullet Normal\left(0,\ \tau_i^2\right)+\left(1-\gamma_i\right)\bullet Normal\left(0,\phi^2\right),\ \phi^2=0.001$,
$\gamma_i\ $~$\ Uniform\ (0,\ 1)$,
$\tau_i^2\ $~$\ Inverse\ Gamma\left(\frac{1}{2},\frac{1}{2}\right)$,
$\beta_0\ $~$\ Cauchy(0,10)$,
$\sigma\ $~$\ {Cauchy}^+(0,\ 10)$
I need to estimate parameters using random-walk Metropolis-Hastings algorithm.
Could anyone tell me the steps? My idea is to sample $\beta_0$, $\beta$ and $\sigma$ from the joint posterior distribution using RW-MH and sample the rest using Gibbs sampler.
I am confused because of Cauchy and half-Cauchy priors on the intercept and sigma, which makes it difficult to sample all parameters using Gibbs sampler as spike-and-slab prior literature usually does.
Or, telling me how to compute the posterior distribution of each parameter would also be appreciated.
Thank you in advance.