Parameter Errors.

Minuit is usually used to find the ``best'' values of a set of parameters, where ``best'' is defined as those values which minimize a given function, FCN. The width of the function minimum, or more generally, the shape of the function in some neighbourhood of the minimum, gives information about the uncertainty in the best parameter values, often called by physicists the parameter errors. An important feature of Minuit is that it offers several tools to analyze the parameter errors. 4cmFCN Normalization and the ERRor definition. Whatever method is used to calculate the parameter errors, they will depend on the overall (multiplicative) normalization of FCN, in the sense that if the value of FCN is everywhere multiplied by a constant $\beta$ , then the errors will be decreased by a factor $\beta$ . Additive constants do not change the parameter errors, but may imply a different goodness-of-fit confidence level.

Assuming that the user knows what the normalization of his FCN means, and also that he is interested in parameter errors, the SET ERRordef command allows him to define what he means by one ``error'', in terms of the change in FCN value which should be caused by changing one parameter by one ``error''. If the FCN is the usual chisquare function (defined below), then [SET ERRordef]ERRordef should be set to 1.0 (the default value anyway) if the user wants the usual one-standard-deviation errors. If FCN is a negative-log-likelihood function, then the one-standard-deviation value for [SET ERRordef]ERRORDEF is 0.5. If FCN is a chisquare, but the user wants two-standard-deviation errors, then [SET ERRordef]ERRORDEF should be = 4.0, etc.

Note that in the usual case where Minuit is being used to perform a fit to some experimental data, the parameter errors will be proportional to the uncertainty in the data, and therefore meaningful parameter errors cannot be obtained unless the measurement errors of the data are known. In the common case of a least-squares fit, FCN is usually defined as a chisquare:

$\chi 2(\alpha )\; =\; \sum$_i=1ⁿ{f(x_i,α) - e_i)²σ_i²}

where $\alpha$ is the vector of free parameters being fitted, and the $\sigma$_i are the uncertainties in the individual measurements $e$_i . If these uncertainties are not known, and are simply left out of the calculation, then the fit may still have meaning, but not the quantitative values of the resulting parameter errors. (Only the relative errors of different parameters with respect to each other may be meaningful.)

If the $\sigma$_i are all overestimated by a factor $\beta$ , then the resulting parameter errors from the fit will be overestimated by the same factor $\beta$ .

• The Error Matrix.
• MINOS Errors.