Ex. 17.5

Ex. 17.5

Consider the setup in Section 17.4.1 with no missing edges. Show that

\[\begin{equation} \bb{S}_{11}\beta - s_{12} = 0 \non \end{equation}\]

are the estimating equations for the multiple regression coefficients of the last variable on the rest.

Soln. 17.5

When no edge is missing, the log-likelihood reduces to the form of (17.11) in the text,

\[\begin{equation} l(\bm{\Theta}) = \log(\text{det}(\bm{\Theta})) - \text{trace}(\bb{S}\bm{\Theta}),\non \end{equation}\]

and \(\bm{\Gamma}=0\) in (17.13). Therefore, following the same arguments which leads to (17.19) in the text, we get

\[\begin{equation} \bb{S}_{11}\beta - s_{12} = 0\non \end{equation}\]

are the estimating equations.