There are no solutions in the frequentist literature to this model when the variance components are unknown. Furthermore, inference is highly sensitive to any assumptions made, due to the low number of observations typically available at the subject level in FMRI.
(11) have proposed an approximate Bayesian solution for the model all-in-one, by assuming that the posterior over the regression parameters is multivariate Normal. However, this does not fully incorporate the full uncertainty of the variance components into the parameters of interest (the regression parameters) at the top level. Indeed, the marginal posterior over the regression parameters turns out to be multivariate t-distributed.
In this section we start by introducing the Bayesian inference framework. However, when using a Bayesian framework, we also need to choose priors for the parameters in our model. In particular, we need to choose priors on the top-level regression and variance parameters. Hence, in the next part of this section we describe how we can use reference priors as noninformative priors.
We could proceed to infer on the full model all-in-one. Instead, by using the fully Bayesian approach with reference priors, we go on to show how we can use summary statistics (from inferring on the first-level model in isolation) as the input into a second level. We show that this gives the same inference as we would obtain from using the full model all-in-one.