Probability content of confidence regions

For an n-parameter problem [MINos]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 , 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]MINOS error in parameter one is the grey area of figure , extending to infinity at both the top and bottom of the figure.

[MINos]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 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 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.

Rectangular confidence region for parameters 1 and 2  

This confidence region for our two-parameter example is the grey area in figure . 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 , 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 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 gives the values of UP which yield hypercontours enclosing given probability contents for given number of parameters.

Optimal confidence region for parameters 1 and 2