Function normalization and ERROR DEF

In order to provide for full generality in the user-defined function value, the user is allowed to define a normalization factor known internally as UP and defined by the Minuit user on an ERROR DEF command card. The default value is one. The Minuit error on a parameter is defined as the change of parameter which would produce a change of the function value equal to UP. This is the most general way to define the error, although in statistics it is more usual to define it in terms of the second derivative of the $\chi^{2}_{}$ function - with respect to the parameter in question. In the simplest linear case (when the function is exactly parabolic at the minimum), the value UP=1.0 corresponds to defining the error as the inverse of the second derivative at the minimum. The fact that Minuit defines the error in terms of a function change does not mean that it always calculates such a function change. Indeed it sometimes (HESSE) calculates the second derivative matrix and inverts it, assuming a parabolic behaviour. This distinction is discussed in section 7.2. The purpose of defining errors by function changes is threefold:

1.

to preserve its meaning in the non-parabolic case (see section 7.2);

2.

to allow generality when the user-defined function is not a chi- square or likelihood, but has some other origin;

3.

to allow calculation not only of ``one-standard deviation'' errors, but also two or more standard deviations, or more general 'confidence regions', especially in the multiparameter case (see section 7.3).

Chi-square normalization

If the user's function value F is supposed to be a chisquare, it must of course be properly normalized. That is, the ``weights'' must in fact correspond to the one-standard-deviation errors on the observations. The most general expression for the chi-square $\chi$ is of the form (see [5], p.163):

$$\chi^{2}_{}$$ = $$\sum_{i,j}^{}$$(x_i - y_i(a))V_ij(x_j - y_j(a))

where x is the vector of observations, y(a) is the vector of fitted values (or theoretical expressions for them) containing the variable fit parameters a, and V is the inverse of the error matrix of the observations x, also known as the covariance matrix of the observations. Fortunately, in most real cases the observations x are statistically independent of each other (e.g., the contents of the bins of a histogram, or measurements of points on a trajectory), so the matrix V is diagonal only. The expression for $\chi^{2}_{}$ then simplifies to the more familiar form:

$$\chi^{2}_{}$$ = $$\sum_{i}^{}$$$${\frac{(x_i - y_i(a))^2}{e_i^2}}$$

where e² is the inverse of the diagonal element of V, the square of the error on the corresponding observation x. In the case where the x are integer numbers of events in an unweighted histogram, for example, the e² are just equal to the x (or to the y, see [5], pp.170-171). The minimization of $\chi^{2}_{}$ above is sometimes called weighted least squares in which case the inverse quantities 1/e² are called the weights. Clearly this is simply a different word for the same thing, but in practice the use of these words sometimes means that the interpretation of e² as variances or squared errors is not straightforward. The word weight often implies that only the relative weights are known (``point two is twice as important as point one'') in which case there is apparently an unknown overall normalization factor. Unfortunately the parameter errors coming out of such a fit will be proportional to this factor, and the user must be aware of this in the formulation of his problem. The e² may also be functions of the fit parameters a (see [5], pp.170-171). Normally this results in somewhat slower convergence of the fit since it usually increases the nonlinearity of the fit. (In the simplest case it turns a linear problem into a non-linear one.) However, the effect on the fitted parameter values and errors should be small. If the user's chi-square function is correctly normalized, he should use UP=1.0 (the default value) to get the usual one standard-deviation errors for the parameters one by one. To get two-standard-dev.eviation errors, use ERROR DEF 4.0, etc., since the chisquare dependance on parameters is quadratic. For more general confidence regions involving more than one parameter, see section 7.2.

Likelihood normalization

If the user function is a negative log-likelihood function, it must again be correctly normalized, but the reasons and ensuing problems in this case are quite different from the chisquare case. The likelihood function takes the form (see [5], p. 155):

F = - $$\sum_{i}^{}$$lnf (x_i, a)

where each x represents in general a vector of observations, the a are the free parameters of the fit, and the function f represents the hypothesis to be fitted. This function f must be normalized:

$$\int$$f (x_i, a)dx₁dx₂...dx_n = constant

that is, the integral of f over all observation space x must be independent of the fit parameters a. The consequence of not normalizing f properly is usually that the fit simply will not converge, some parameters running away to infinity. Strangely enough, the value of the normalization constant does not affect the fitted parameter values or errors, as can be seen by the fact that the logarithm makes a multiplicative constant into an additive one, which simply shifts the whole log-likelihood curve and affects its value, but not the fitted parameter values or errors. In fact, the actual value of the likelihood at the minimum is quite meaningless (unlike the chi-square value) and even depends on the units in which the observation space x is expressed. The meaningful quantity is the difference in log-likelihood between two points in parameter-space, which is dimensionless. For likelihood fits, the value UP=0.5 corresponds to one-standard-deviation errors. Or, alternatively, F may be defined as -2log(likelihood), in which case differences in F have the same meaning as for chi-square and UP=1.0 is appropriate. The two different ways of introducing the factor of 2 are quite equivalent in Minuit, and although most people seem to use UP=0.5, it is perhaps more logical to put the factor 2 directly into FCN.