Full hierarchical model excluding the power spectrum hyper-priors for the fields. The lowest layer contains the data sets d_ff and d_ϕ, which are connected by Eqs. (4) and (16) to the sky maps EM_ff and ϕ. The sky maps in turn are connected to the Gaussian fields in the uppermost layer via the respective models defined in Eqs. (18) and (20). From there on, the higher branches for the respective correlation structure inference follow. They are not depicted here, but are shown and explained in the appendix.