Today I managed to finish the Metropolis-within-Gibbs MCMC code. In principle, this code can be used to several hierarchical Bayesian problems we have open at the moment. All of them share the property that the posterior depends on a large set of parameters $\theta_i$ which have a common prior that also depends on hyperparameters. The advantage is that the influence of each $\theta_i$ on the posterior is independent of the other set of $\theta_j$ with $j \neq i$.

The first problem we have is the one of detecting magnetic fields in central stars of planetary nebulae when the noise variance is left as an unknown. The likelihood is then transformed into a Student-t distribution. In this case, the marginal posterior for the hyperparameters cannot be found in a closed form as it happens when the noise variance is known. My guess is that a Metropolis-within-Gibbs can be used to rapidly sample from the posterior. Other problems include obtaining global information about the physical parameters of coronal loop oscillations and also about the magnetism of the quiet Sun.

On a short discussion with rms, he suggested to develop a Bayesian meta analysis of some atomic data in astrophysics. This is a really good idea that can be solved within a hierarchical Bayesian model. We will work on it in the following weeks.