The Reliability of Minuit Error Estimates.

Minuit always carries around its own current estimates of the parameter errors, which it will print out on request, no matter how accurate they are at any given point in the execution. For example, at initialization, these estimates are just the starting step sizes as specified by the user. After a [MIGrad]MIGRAD or [HESse]HESSE step, the errors are usually quite accurate, unless there has been a problem. Minuit, when it prints out error values, also gives some indication of how reliable it thinks they are. For example, those marked 'CURRENT GUESS ERROR' are only working values not to be believed, and 'APPROXIMATE ERROR' means that they have been calculated but there is reason to believe that they may not be accurate. If no mitigating adjective is given, then at least Minuit believes the errors are accurate, although there is always a small chance that Minuit has been fooled. Some visible signs that Minuit may have been fooled are:

• Warning messages produced during the minimization or error analysis.
• Failure to find new minimum.
• Value of EDM too big. For a ``normal'' minimization, after [MIGrad]MIGRAD, the value of EDM is usually more than three orders of magnitude smaller than UP (the SET ERRordef), unless a looser tolerance has been specified.
• Correlation coefficients exactly equal to zero, unless some parameters are known to be uncorrelated with the others.
• Correlation coefficients very close to one (greater than 0.99). This indicates both an exceptionally difficult problem, and one which has been badly parametrized so that individual errors are not very meaningful because they are so highly correlated.
• Parameter at limit. This condition, signalled by a Minuit warning message, may make both the function minimum and parameter errors unreliable. See section , Getting the right parameter errors with limits.

The best way to be absolutely sure of the errors, is to use ``independent'' calculations and compare them, or compare the calculated errors with a picture of the function if possible. For example, if there is only one free parameter, the command [SCAn]SCAN allows the user to verify approximately the function curvature. Similarly, if there are only two free parameters, use [CONtour]CONTOUR. To verify a full error matrix, compare the results of [MIGrad]MIGRAD with those (calculated afterward) by [HESse]HESSE, which uses a different method. And of course the most reliable and most expensive technique, which must be used if asymmetric errors are required, is [MINOs]MINOS.

4cmConvergence in MIGRAD, and Positive-definiteness.

[MIGrad]MIGRAD uses its current estimate of the covariance matrix of the function to determine the current search direction, since this is the optimal strategy for quadratic functions and ``physical'' functions should be quadratic in the neighbourhood of the minimum at least. The search directions determined by [MIGrad]MIGRAD are guaranteed to be downhill only if the covariance matrix is positive-definite, so in case this is not true, it makes a positive-definite approximation by adding an appropriate constant along the diagonal as determined by the eigenvalues of the matrix. Theoretically, the covariance matrix for a ``physical'' function must be positive-definite at the minimum, although it may not be so for all points far away from the minimum, even for a well-determined physical problem. Therefore, if [MIGrad]MIGRAD reports that it has found a non-positive-definite covariance matrix, this may be a sign of one or more of the following:

• A non-physical region. On its way to the minimum, [MIGrad]MIGRAD may have traversed a region which has unphysical behaviour, which is of course not a serious problem as long as it recovers and leaves such a region.
• An underdetermined problem. If the matrix is not positive-definite even at the minimum, this may mean that the solution is not well-defined, for example that there are more unknowns than there are data points, or that the parametrization of the fit contains a linear dependence. If this is the case, then Minuit (or any other program) cannot solve your problem uniquely, and the error matrix will necessarily be largely meaningless, so the user must remove the underdeterminedness by reformulating the parametrization. Minuit cannot do this itself, but it can provide some hints (contours, global correlation coefficients, eigenvalues) which can help the clever user to find out what is wrong.
• Numerical inaccuracies. It is possible that the apparent lack of positive-definiteness is in fact only due to excessive roundoff errors in numerical calculations, either in FCN or in Minuit. This is unlikely in general, but becomes more likely if the number of free parameters is very large, or if the parameters are badly scaled (not all of the same order of magnitude), and correlations are also large. In any case, whether the non-positive-definiteness is real or only numerical is largely irrelevant, since in both cases the error matrix will be unreliable and the minimum suspicious.