[MINos]MINOS is designed to calculate the correct errors in all cases, especially when there are non-linearities as described above. The theory behind the method is described in [5], pp. 204-205 (where ``non-parabolic likelihood'' should of course read ``non-parabolic log-likelihood'', which is equivalent to ``nonparabolic chi-square'').
[MINos]MINOS actually follows the function out from the minimum to find
where it crosses the function value (minimum + UP
), instead of using
the curvature at the minimum and assuming a parabolic shape. This
method not only yields errors which may be different from those of
[HESse]HESSE, but in general also different positive and negative errors
(asymmetric error interval). Indeed the most frequent result for
most physical problems is that the (symmetric) [HESse]HESSE error lies
between the positive and negative errors of [MINos]MINOS. The difference
between these three numbers is one measure of the non-linearity of
the problem (or rather of its formulation).
In practice, [MINos]MINOS errors usually turn out to be close to, or
somewhat larger than errors derived from the error matrix, although
in cases of very bad behaviour (very little data or ill-posed model)
anything can happen. In particular, it is often not
true in [MINos]MINOS that two-standard-deviation errors
(UP=4
) and three-standard-deviation errors (UP=9
)
are respectively two and three times as big as one-standard-deviation errors,
as is true by definition for errors derived from the
error matrix ([MIGrad]MIGRAD or [HESse]HESSE).