Hierarchical priors

It is often desirable that the priors of the parameters should not be too restrictive. A common type of a vague prior is the hierarchical prior Gelman95. For example the priors of the elements $a_{ij}$ of a mixing matrix ${\mathbf{A}}$ can be defined via the Gaussian distributions

$$p( a_{ij} \mid v^a_i ) = \mathcal N( a_{ij} ; 0, \exp(-v^a_i) )$$ (49)

$$p( v^a_i \mid m^{va}, v^{va} ) = \mathcal N( v^a_i ; m^{va}, \exp(-v^{va}) ) \, .$$ (50)

Finally, the priors of the quantities $m^{va}$ and $v^{va}$ have flat Gaussian distributions $\mathcal N(\cdot ; 0,100)$ (the constants depending on the scale of the data). When going up in the hierarchy, we use the same distribution for each column of a matrix and for each component of a vector. On the top, the number of required constant priors is small. Thus very little information is provided and needed a priori. This kind of hierarchical priors are used in the experiments later on this paper.