probability of having a reduced data set, $D_{\rm reduced} =
\{B_n\}$, given the set of model parameters, ${{\mit \Theta}}$, is therefore given by $$P(D_{\rm reduced}|{{\mit \Theta}}) =
   \frac{1}{\sqrt{(2\pi)^M \det C({{\mit \Theta}})}}
   \exp\left[ -\frac12
     \left({{\mbox{\boldmath $B$}}}^{\rm T} - \langle{{\mbox{\boldmath $B$}}}^{\rm T}\rangle_{{{\mit \Theta}}}\right)
       C^{-1}({{\mit \Theta}}) \left({{\mbox{\boldmath $B$}}} -
   \langle{{\mbox{\boldmath $B$}}}\rangle_{{{\mit \Theta}}}\right)
   \right].
\label{eq2-0}$$ The above equation assumes that the procedure of obtaining data is independent of the assumed model parameters, so that the projection matrix should be fixed throughout analyzing and maximizing the likelihood function. However, the optimal choice of the projection matrix usually depends on the underlying correlation function, and thus on model parameters, as explicitly presented in Appendix \[app1\] in the case of the KL-mode projection. Therefore, the projection matrix is initially constructed by a fiducial set of model parameters, which is fixed throughout the maximization of the likelihood function.

While the reduced correlation matrix $C$ constructed by KL eigenmodes is diagonalized for a fiducial set of model parameters, it should be noted that the correlation matrix $C$ is [*not*]{} diagonal for other set of model parameters. Therefore, the inversion of the correlation matrix $C$ still needs to be performed. The likelihood function is not biased by the initial choice 