672
L. Lyons
that was genuinely more precise than expected 19 How to deal with this situation in
general is an open question. It has features in common with the problem (inspired by
ref. [55]) of measuring a voltage by choosing at random a voltmeter from a cupboard
containing meters of different sensitivities.
Another example involves combining two measurements of a cross-section
with small statistical uncertainties, but with large correlated uncertainties from the
common luminosity. With this luminosity uncertainty included in the covariance
matrix, BLUE can result in the combined value being outside the range of the
individual measurements. For this situation, it is preferable to exclude the luminosity
uncertainty from the covariance matrix, and to apply it to the combined result
afterwards.
15.5.2 Two or More Parameters
An extension of this procedure is for combining N pairs of correlated measurements
(e.g. the gradient and intercept of a straight line fit to several sets of data, where for
simplicity it is assumed that any pair is independent of every other pair). For several
pairs of values (a i , b i ) with inverse covariance matrices M i , the best combined
values (a comb , b comb ) have as their inverse covariance matrix M = i . This
means that, if the covariance matrix correlation coefficients ρ i of the different
measurements are very different from each other, the uncertainty on a comb can be
much smaller than that for any single measurement.
This situation applies for track fitting to hits in a series of groups of tracking
chambers, where each set of close chambers provides a very poor determination
of the track; but the combination involves widely spaced chambers and determines
the track well. Using the profile likelihoods (e.g. for the intercept, profiled over the
gradient) for combining different measurements loses the correlation information
and can lead to a very poor combined estimate[37]. The alternative of ignoring the
correlation information is also strongly discouraged.
The importance of retaining covariances is relevant for many combinations, e.g.
for the determination of the amount of Dark Energy in the Universe from various
cosmological data[73].
19 For example, the ALEPH experiment at LEP produced a tighter-than-expected upper limit on
the mass of ν τ because they happened to observe τ decay configurations which were particularly
sensitive to the ν τ mass.
L. Lyons
that was genuinely more precise than expected 19 How to deal with this situation in
general is an open question. It has features in common with the problem (inspired by
ref. [55]) of measuring a voltage by choosing at random a voltmeter from a cupboard
containing meters of different sensitivities.
Another example involves combining two measurements of a cross-section
with small statistical uncertainties, but with large correlated uncertainties from the
common luminosity. With this luminosity uncertainty included in the covariance
matrix, BLUE can result in the combined value being outside the range of the
individual measurements. For this situation, it is preferable to exclude the luminosity
uncertainty from the covariance matrix, and to apply it to the combined result
afterwards.
15.5.2 Two or More Parameters
An extension of this procedure is for combining N pairs of correlated measurements
(e.g. the gradient and intercept of a straight line fit to several sets of data, where for
simplicity it is assumed that any pair is independent of every other pair). For several
pairs of values (a i , b i ) with inverse covariance matrices M i , the best combined
values (a comb , b comb ) have as their inverse covariance matrix M = i . This
means that, if the covariance matrix correlation coefficients ρ i of the different
measurements are very different from each other, the uncertainty on a comb can be
much smaller than that for any single measurement.
This situation applies for track fitting to hits in a series of groups of tracking
chambers, where each set of close chambers provides a very poor determination
of the track; but the combination involves widely spaced chambers and determines
the track well. Using the profile likelihoods (e.g. for the intercept, profiled over the
gradient) for combining different measurements loses the correlation information
and can lead to a very poor combined estimate[37]. The alternative of ignoring the
correlation information is also strongly discouraged.
The importance of retaining covariances is relevant for many combinations, e.g.
for the determination of the amount of Dark Energy in the Universe from various
cosmological data[73].
19 For example, the ALEPH experiment at LEP produced a tighter-than-expected upper limit on
the mass of ν τ because they happened to observe τ decay configurations which were particularly
sensitive to the ν τ mass.