Multiparameter errors

In addition to the difficulties described above, a special class of problems arise in interpreting errors when there is more than one free parameter. These problems are quite separate from those described above and are really much simpler in principle, although in practice confusion often arises.

The Error Matrix

The error matrix, also called the covariance matrix, is the inverse of the second derivative matrix of the (log-likelihood or chisquare) function with respect to its free parameters, usually assumed to be evaluated at the best parameter values (the function minimum). The diagonal elements of the error matrix are the squares of the individual parameter errors, including the effects of correlations with the other parameters. The inverse of the error matrix, the second derivative matrix, has as diagonal elements the second partial derivatives with respect to one parameter at a time. These diagonal elements are not therefore coupled to any other parameters, but when the matrix is inverted, the diagonal elements of the inverse contain contributions from all the elements of the second derivative matrix, which is ``where the correlations come from''. Although a parameter may be either positively or negatively correlated with another, the effect of correlations is always to increase the errors on the other parameters in the sense that if a given free parameter suddenly became exactly known (fixed), that would always decrease (or at least not change) the errors on the other parameters. In order to see this effect quantitatively, the following procedure can be used to ``delete'' one parameter from the error matrix, including its effects on the other parameters:

1.

Invert the error matrix, to yield the second-derivative matrix.

2.

Remove the row and column of the inverse corresponding to the given parameter.

3.

Re-invert the resulting (smaller) matrix.

This reduced error matrix will have its diagonal elements smaller or equal to the corresponding elements in the original error matrix, the difference representing the effect of knowing or not knowing the true value of the parameter that was removed at step two. This procedure is exactly that performed by Minuit when a FIX command is executed. Note that it is not reversible, since information has been lost in the deletion. The Minuit commands RESTORE and RELEASE therefore cause the error matrix to be considered lost and it must be recalculated entirely.

MINOS with several free Parameters

The MINOS algorithm is described in some detail in part 1 of this manual. Here we add some supplementary ``geometrical interpretation'' for the multidimensional case. Let us consider that there are just two free parameters, and draw the contour line connecting all points where the function takes on the value F_min + UP. (The CONTOUR command will do this for you from Minuit). For a linear problem, this contour line would be an exact ellipse, the shape and orientation of which are described in [5], p.196 (Fig. 9.4). For our problem let the contour be as in figure 7.1. If MINOS is requested to find the errors in parameter one (the x-axis), it will find the extreme contour points A and B, whose x-coordinates, relative to the x-coordinate at the minimum (X), will be respectively the negative and positive MINOS errors of parameter one.

Probability content of confidence regions

For an n-parameter problem MINOS performs minimizations in (n - 1) dimensions in order to find the extreme points of the hypercontour of which a two-dimensional example is given in figure 7.1, and in this way takes account of all the correlations with the other n - 1 parameters. However, the errors which it calculates are still only single-parameter errors, in the sense that each parameter error is a statement only about the value of that parameter. This is represented geometrically by saying that the confidence region expressed by the MINOS error in parameter one is the grey area of figure 7.2, extending to infinity at both the top and bottom of the figure.

Figure 7.1: MINOS errors for parameter 1

Figure 7.2: MINOS error confidence region for parameter 1

If UP is set to the appropriate one-standard-deviation value, then the precise meaning of the confidence region of figure 7.2 is: ``The probability that the true value of parameter one lies between A and B is 68.3%'' (the probability of a normally-distributed parameter lying within one std.-dev. of its mean). That is, the probability content of the grey area in figure 7.2 is 68.3%. No statement is made about the simultaneous values of the other parameter(s), since the grey area covers all values of the other parameter(s). If it is desired to make simultaneously statements about the values of two or more parameters, the situation becomes considerably more complicated and the probabilities get much smaller. The first problem is that of choosing the shape of the confidence region, since it is no longer simply an interval on an axis, but a hypervolume. The easiest shape to express is the hyperrectangle given by:

A < param 1 < B
C < param 2 < D
E < param 3 < F, etc.

This confidence region for our two-parameter example is the grey area in figure 7.3. However, there are two good reasons not to use such a shape:

1.

Some regions inside the hyperrectangle (namely the corners) have low likelihoods, lower than some regions just outside the rectangle, so the hyperrectangle is not the optimal shape (does not contain the most likely points).

2.

One does not know an easy way to calculate the probability content of these hyperrectangles (see [5], p.196-197, especially fig. 9.5a).

For these reasons one usually chooses regions delimited by contours of equal likelihood (hyperellipsoids in the linear case). For our two-parameter example, such a confidence region would be the grey region in figure 7.4, and the corresponding probability statement is: ``The probability that parameter one and parameter two simultaneously take on values within the one-standard-deviation likelihood contour is 39.3%''. The probability content of confidence regions like those shaded in figure 7.4 becomes very small as the number of parameters NPAR increases, for a given value of UP. Such probability contents are in fact the probabilities of exceeding the value UP for a chisquare function of NPAR degrees of freedom, and can therefore be read off from tables of chisquare. Table 7.1 gives the values of UP which yield hypercontours enclosing given probability contents for given number of parameters.

Figure 7.3: Rectangular confidence region for parameters 1 and 2

Figure 7.4: Optimal confidence region for parameters 1 and 2

Table 7.1: Table of UP for multi-parameter confidence regions

	Confidence level
Nb. parameters	(probability contents desired inside hypercontour $\chi^{2}_{}$ = $\chi^{2}_{\mathrm{min}}$ + UP)
	50%	70%	90%	95%	99%
1	0.46	1.07	2.70	3.84	6.63
2	1.39	2.41	4.61	5.99	9.21
3	2.37	3.67	6.25	7.82	11.36
4	3.36	4.88	7.78	9.49	13.28
5	4.35	6.06	9.24	11.07	15.09
6	5.35	7.23	10.65	12.59	16.81
7	6.35	8.38	12.02	14.07	18.49
8	7.34	9.52	13.36	15.51	20.09
9	8.34	10.66	14.68	16.92	21.67
10	9.34	11.78	15.99	18.31	23.21
11	10.34	12.88	17.29	19.68	24.71
	For FCN -log(likelihood) instead of $\chi^{2}_{}$, all values of UP to be divided by 2.