MCMC

Here, we use Markov Chain Monte Carlo (MCMC) to sample from the full joint posterior distribution given in equation 14. This also automatically provides us with samples from the marginal posterior distribution, $p(\beta_g\vert Y)$.

We use single-component Metropolis-Hastings jumps (i.e. we propose separate jumps for each of the parameters in turn) for all parameters. We use separate Normal proposal distributions for each parameter, with the mean fixed on the current value, and with a scale parameter $\sigma_p$ for the $p^{th}$ parameter that is updated every 30 jumps. At the $j^{th}$ update $\sigma_p$ is updated according to:

$$\sigma_p^{j+1}=\sigma_p^{j}\tilde{R}\frac{(1+A+R)}{(1+R)}$$ (19)

where $A$ and $R$ are the number of accepted and rejected jumps since the last $\sigma_p$ update respectively, $\tilde{R}$ is the desired rejection rate, which we fix at $0.5$.

We require a good initialisation of the parameters in the model purely to reduce the required burn-in of the MCMC chains (the burn-in is the part of the MCMC chain which is used to ensure that the chain has converged to be sampling from the true distribution). To initialise we use the fast approximation approach described in section 3.5.