  We discuss the effect of large positive correlations in the combinations of
several measurements of a single physical quantity using the Best Linear
Unbiased Estimate (BLUE) method. We suggest a new approach for comparing the
relative weights of the different measurements in their contributions to the
combined knowledge about the unknown parameter, using the well-established
concept of Fisher information. We argue, in particular, that one We discuss the
effect of large positive correlations in the combinations of several
measurements of a single physical quantity using the Best Linear Unbiased
Estimate (BLUE) method. We suggest a new approach for comparing the relative
weights of the different measurements in their contributions to the combined
knowledge about the unknown parameter, using the well-established concept of
Fisher information. We argue, in particular, that one contribution to
information comes from the collective interplay of the measurements through
their correlations and that this contribution cannot be attributed to any of
the individual measurements alone. We show that negative coefficients in the
BLUE weighted average invariably indicate the presence of a regime of high
correlations, where the effect of further increasing some of these correlations
is that of reducing the error on the combined estimate. In these regimes, we
stress that assuming fully correlated systematic uncertainties is not a truly
conservative choice, and that the correlations provided as input to BLUE
combinations need to be assessed with extreme care instead. In situations where
the precise evaluation of these correlations is impractical, or even
impossible, we provide tools to help experimental physicists perform more
conservative combinations.
