Posterior propriety of an objective prior for generalized hierarchical normal linear models

Figures & data

Table 1. Summary of certain important notations within model (3(3) $\{\begin{array}{lll}\mathrm{L}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l}1:& \begin{array}{l}{\mathit{y}}_{i_0}={\mathit{Z}}_{0i_0}{\mathit{\theta }}_1\\ +{\mathcal{N}}_{k_0}(\mathbf{0},{\mathit{I}}_{k_0}),\end{array}& \begin{array}{l}{i}_0\in \left[m_0\right],\\ {\mathit{\theta }}_1^{\mathrm{\top }}=({\mathit{\theta }}_{11}^{\mathrm{\top }},\\ \dots ,{\mathit{\theta }}_{1m_1}^{\mathrm{\top }});\end{array}\\ \mathrm{L}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l}2:& \begin{array}{l}{\mathit{\theta }}_{1i_1}={\mathit{Z}}_{1i_1}{\mathit{\theta }}_2\\ +{\mathcal{N}}_{k_1}(\mathbf{0},{\mathit{V}}_1),\end{array}& \begin{array}{l}{i}_1\in \left[m_1\right],{\mathit{\theta }}_2^{\mathrm{\top }}\\ =({\mathit{\theta }}_{21}^{\mathrm{\top }},\\ \dots ,{\mathit{\theta }}_{2m_2}^{\mathrm{\top }});\end{array}\\ ⋮& ⋮& ⋮\\ \mathrm{L}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l}r:& \begin{array}{l}{\mathit{\theta }}_{r-1,i_{r-1}}\\ ={\mathit{Z}}_{r-1,i_{r-1}}{\mathit{\theta }}_r\\ +{\mathcal{N}}_{k_{r-1}}(\mathbf{0},{\mathit{V}}_{r-1}),\end{array}& \begin{array}{l}{i}_{r-1}\in \left[m_{r-1}\right],\\ {\mathit{\theta }}_r^{\mathrm{\top }}=({\mathit{\theta }}_{r1}^{\mathrm{\top }},\\ \dots ,{\mathit{\theta }}_{r{m}_r}^{\mathrm{\top }});\end{array}\\ \mathrm{L}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l}r+1:& \begin{array}{l}{\mathit{\theta }}_{r{i}_r}={\mathit{Z}}_{r{i}_r}\mathit{\eta }\\ +{\mathcal{N}}_{k_r}(\mathbf{0},{\mathit{V}}_r),\end{array}& i_r\in \left[m_r\right].\end{array}$(3) ) and $j=0,1,\dots ,r$.

Notation	Meaning
r + 1	Total number of levels
$m_j$	Number of the output units in level $(j+1)$
$k_j$	Dimension of each output unit in level $(j+1)$
d	Dimension of the fixed effect $\mathit{\eta }$