We can now use our original adaptive MCMC algorithm to sample from the bottom level parameters, combined with classical Gibbs sampling and Eq. 21 to sample from the hyperparameters (see Algorithm 1 for details). For the hyperprior parameters we use κ 0 = 0 and ν 0 = 1. We set µ 0 to be the centre point of the 5-dimensional hypercube in Eq. 14, and Ψ as a diagonal matrix with the standard deviation of each parameter set to half the width of this same hypercube. To prevent the bottom level MCMC chains from exploring unphysical parameter regimes, we automatically reject any proposed point which lies outside of the hypercube.