BIDET

MCMC can be used to directly obtain samples from $p(\beta_g\vert Y)$. However, we would need to get lots of samples well into the tail of the distribution, and MCMC sampling is computationally intensive. Hence, we avoid the need for large numbers of samples by assuming that $p(\beta_g\vert Y)$ is a multivariate non-central t-distribution. Recall that assuming a multivariate non-central t-distribution is also important to the idea of being able to split hierarchies into inference on different levels. Therefore, we clean up the samples of the posterior using Bayesian Inference with Distribution Estimation using a T-fit (BIDET).

BIDET fits a multivariate non-central t-distribution to the MCMC samples of $p(\beta_g\vert Y)$ as described in appendix 10.4. Figure 1 shows the result of using the multivariate non-central t-distribution fit to an MCMC chain obtained (see section 3.6) on a voxel in Dataset 2 described in section 6.

Figure 1: The (in this case 1-dimensional) t-fit obtained on the MCMC samples from a voxel in dataset 1.