Several times already in this chapter we have made statements about the standard errors, or uncertainties, in a set of M estimated parameters a. We have given some formulas for computing standard deviations or variances of individual parameters | Confidence Limits on Estimated Model Parameters 689 Confidence Limits on Estimated Model Parameters Several times already in this chapter we have made statements about the standard errors or uncertainties in a set of M estimated parameters a. We have given some s j o formulas for computing standard deviations or variances of individual parameters equations as well as some formulas for covariances Jp-j between pairs of parameters equation remark following equation g Q equation equation . Os j In this section we want to be more explicit regarding the precise meaning of these quantitative uncertainties and to give further information about how quantitative confidence limits on fitted parameters can be estimated. The subject 8 o o can get somewhat technical and even somewhat confusing so we will try to make precise statements even when they must be offered without proof. Figure shows the conceptual scheme of an experiment that measures - c a set of parameters. There is some underlying true set of parameters atrue that are g known to Mother Nature but hidden from the experimenter. These true parameters are statistically realized along with random measurement errors as a measured data set which we will symbolize as D 0 . The data set D 0 is known to the experimenter. 1 He or she fits the data to a model by 2 minimization or some other technique and g. k obtains measured . fitted values for the parameters which we here denote a 0 . Because measurement errors have a random component D 0 is not a unique H realization of the true parameters atrue. Rather there are infinitely many other g. 00 realizations of the true parameters as hypothetical data sets each of which could . have been the one measured but happened not to be. Let us symbolize these by D i D 2 . . . Each one had it been realized would have given a slightly different set of fitted parameters a 1 a 2 . respectively. These parameter sets g
