Figure 3. Example of estimation under a mixed-effects or hierarchical linear model.

The inversion was cross-validated with expectation maximization (EM), where the M-step corresponds to restricted maximum likelihood (ReML). This example used a simple two-level model that embodies empirical shrinkage priors on the first-level parameters. These models are also known as parametric empirical Bayes (PEB) models (left). Causes were sampled from the unit normal density to generate a response, which was used to recover the causes, given the parameters. Slight differences in the hyperparameter estimates (upper right), due to a different hyperparameterisation, have little effect on the conditional means of the unknown causes (lower right), which are almost indistinguishable.