Interpretation of Parameter Errors

There are two kinds of problems that can arise: The reliability of Minuit's error estimates, and their statistical interpretation, assuming they are accurate.

Statistical Interpretation.

For discussuion of basic concepts, such as the meaning of the elements of the error matrix, parabolic versus MINOS errors, the appropriate value for UP (see SET ERRdef), and setting of exact confidence levels, see (in order of increasing complexity and completeness):

• ``Interpretation of the Errors on Parameters'', see Part 3 of this write-up.
• ``Determining the Statistical Significance of Experimental Results''[4].
• ``Statistical Methods in Experimental Physics''[5].

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 or 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, 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 5.3.2, 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 allows the user to verify approximately the function curvature. Similarly, if there are only two free parameters, use CONTOUR. To verify a full error matrix, compare the results of MIGRAD with those (calculated afterward) by 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.