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\begin_body

\begin_layout Section*
Conditional Sensitivity
\end_layout

\begin_layout Standard
Suppose 
\begin_inset Formula $\hat{\theta}_{inc}$
\end_inset

 is a vector containing subset of parameters 
\begin_inset Formula $\hat{\theta}$
\end_inset

 and 
\begin_inset Formula $\hat{\theta}_{excl}$
\end_inset

 is a vector containing the remaining parameters.
 Given the full covariance matrix over 
\begin_inset Formula $(\hat{\theta},\hat{\gamma})$
\end_inset

 as defined in the paper, we can reorder the rows and columns such that
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\Sigma=\left(\begin{array}{cc}
\Sigma_{11} & \Sigma_{12}\\
\Sigma_{21} & \Sigma_{22}
\end{array}\right)
\]

\end_inset

where 
\begin_inset Formula $\Sigma_{11}$
\end_inset

 is variance matrix of 
\begin_inset Formula $(\hat{\theta}_{inc},\hat{\gamma})$
\end_inset

, 
\begin_inset Formula $\Sigma_{22}$
\end_inset

 is variance matrix of 
\begin_inset Formula $\hat{\theta}_{excl}$
\end_inset

, and 
\begin_inset Formula $\Sigma_{12}$
\end_inset

 is the covariance of 
\begin_inset Formula $(\hat{\theta}_{inc},\hat{\gamma})$
\end_inset

 and 
\begin_inset Formula $\hat{\theta}_{excl}$
\end_inset

.
 The conditional covariance of 
\begin_inset Formula $(\hat{\theta}_{inc},\hat{\gamma})$
\end_inset

 conditioned on 
\begin_inset Formula $\hat{\theta}_{excl}$
\end_inset

 is then given by:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\bar{\Sigma}=\Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}
\]

\end_inset

Then the sensitivity of 
\begin_inset Formula $\hat{\theta}_{inc}$
\end_inset

 to 
\begin_inset Formula $\hat{\gamma}$
\end_inset

 conditional on 
\begin_inset Formula $\hat{\theta}_{excl}$
\end_inset

 is 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\bar{\Lambda}=\bar{\Sigma}_{\theta_{inc}\gamma}\bar{\Sigma}_{\gamma\gamma}^{-1}
\]

\end_inset


\end_layout

\begin_layout Section*
Transform Sensitivity
\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $A$
\end_inset

 be a linear operator on 
\begin_inset Formula $\hat{\gamma}$
\end_inset

 represented as a matrix.
 Define 
\begin_inset Formula $\Lambda_{A}$
\end_inset

 to be the sensitivity of 
\begin_inset Formula $\hat{\theta}$
\end_inset

 to 
\begin_inset Formula $A\hat{\gamma}$
\end_inset

.
 Observe that
\begin_inset Formula 
\[
\Cov\left(\theta,\gamma\right)=\left(\begin{array}{cc}
\Sigma_{\theta\theta} & \Sigma_{\theta\gamma}A'\\
A\Sigma_{\gamma\theta} & A\Sigma_{\gamma\gamma}A'
\end{array}\right)
\]

\end_inset

and therefore by definition
\begin_inset Formula 
\[
\Lambda_{A}=\Sigma_{\theta\gamma}A'\left(A\Sigma_{\gamma\gamma}A'\right)^{-1}
\]

\end_inset

To compute standardized transform sensitivity we would simply scale by the square
 root of the diagonals of 
\begin_inset Formula $\Sigma_{\theta\theta}$
\end_inset

 and 
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\begin_inset Formula $\left(A\Sigma_{\gamma\gamma}A'\right)^{-1}$
\end_inset

 by analogy with standardized sensitivity as defined in the paper.
\end_layout

\end_body
\end_document
