  We address a frequently asked question on the covariance fitting of the
highly correlated data such as our $B_K$ data based on the SU(2) staggered
chiral perturbation theory. Basically, the essence of the problem is that we do
not have an accurate fitting function enough to fit extremely precise data.
When eigenvalues of the covariance matrix are small, even a tiny error of
fitting function yields large chi-square and spoils the fitting procedure. We
have applied a number of prescriptions available in the market such as the
cut-off method, modified covariance matrix method, and Bayesian method. We also
propose a brand new method, the eigenmode shift method which allows a full
covariance fitting without modifying the covariance matrix at all. In our case,
the eigenmode shift (ES) method and Bayesian method turn out to be the best
prescription to the problem. We also provide a pedagogical example of data
analysis in which the diagonal approximation and the cut-off method fail in
fitting manifestly, but the ES method and the Bayesian approach work well.
