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15.6 Confidence Limits on Estimated Model Parameters68915.6 Confidence Limits on Estimated ModelParametersMonte Carlo Simulation of Synthetic Data SetsAlthough the measured parameter set a(0) is not the true one, let us considera fictitious world in which it was the true one. Since we hope that our measuredparameters are not too wrong, we hope that that fictitious world is not too differentfrom the actual world with parameters atrue . In particular, let us hope — no, let usassume — that the shape of the probability distribution a(i) − a(0) in the fictitiousworld is the same, or very nearly the same, as the shape of the probability distributionSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).Several times already in this chapter we have made statements about the standarderrors, or uncertainties, in a set of M estimated parameters a. We have given someformulas for computing standard deviations or variances of individual parameters(equations 15.2.9, 15.4.15, 15.4.19), as well as some formulas for covariancesbetween pairs of parameters (equation 15.2.10; remark following equation 15.4.15;equation 15.4.20; equation 15.5.15).In this section, we want to be more explicit regarding the precise meaningof these quantitative uncertainties, and to give further information about howquantitative confidence limits on fitted parameters can be estimated.
The subjectcan get somewhat technical, and even somewhat confusing, so we will try to makeprecise statements, even when they must be offered without proof.Figure 15.6.1 shows the conceptual scheme of an experiment that “measures”a set of parameters. There is some underlying true set of parameters atrue that areknown to Mother Nature but hidden from the experimenter. These true parametersare statistically realized, along with random measurement errors, as a measured dataset, which we will symbolize as D(0). The data set D(0) is known to the experimenter.He or she fits the data to a model by χ2 minimization or some other technique, andobtains measured, i.e., fitted, values for the parameters, which we here denote a(0) .Because measurement errors have a random component, D(0) is not a uniquerealization of the true parameters atrue .
Rather, there are infinitely many otherrealizations of the true parameters as “hypothetical data sets” each of which couldhave been the one measured, but happened not to be. Let us symbolize theseby D(1) , D(2), . . . . Each one, had it been realized, would have given a slightlydifferent set of fitted parameters, a(1), a(2), . . . , respectively. These parameter setsa(i) therefore occur with some probability distribution in the M -dimensional spaceof all possible parameter sets a. The actual measured set a(0) is one member drawnfrom this distribution.Even more interesting than the probability distribution of a(i) would be thedistribution of the difference a(i) − atrue . This distribution differs from the formerone by a translation that puts Mother Nature’s true value at the origin.
If we knew thisdistribution, we would know everything that there is to know about the quantitativeuncertainties in our experimental measurement a(0) .So the name of the game is to find some way of estimating or approximatingthe probability distribution of a(i) − atrue without knowing atrue and without havingavailable to us an infinite universe of hypothetical data sets.690Modeling of Dataactual data setχ2minfittedparametersa0peexhypotheticaldata seta1hypotheticaldata seta2hypotheticaldata seta3......true parametersa trueFigure 15.6.1. A statistical universe of data sets from an underlying model.
True parameters a true arerealized in a data set, from which fitted (observed) parameters a 0 are obtained. If the experiment wererepeated many times, new data sets and new values of the fitted parameters would be obtained.a(i) − atrue in the real world. Notice that we are not assuming that a(0) and atrue areequal; they are certainly not. We are only assuming that the way in which randomerrors enter the experiment and data analysis does not vary rapidly as a function ofatrue , so that a(0) can serve as a reasonable surrogate.Now, often, the distribution of a(i) − a(0) in the fictitious world is within ourpower to calculate (see Figure 15.6.2).
If we know something about the processthat generated our data, given an assumed set of parameters a(0) , then we canusually figure out how to simulate our own sets of “synthetic” realizations of theseparameters as “synthetic data sets.” The procedure is to draw random numbers fromappropriate distributions (cf. §7.2–§7.3) so as to mimic our best understanding ofthe underlying process and measurement errors in our apparatus. With such randomdraws, we construct data sets with exactly the same numbers of measured points,and precisely the same values of all control (independent) variables, as our actualSS, D(2), .
. . . By constructiondata set D(0) . Let us call these simulated data sets D(1)these are supposed to have exactly the same statistical relationship to a(0) as theD(i) ’s have to atrue . (For the case where you don’t know enough about what youare measuring to do a credible job of simulating it, see below.)S, perform exactly the same procedure for estimation ofNext, for each D(j)2parameters, e.g., χ minimization, as was performed on the actual data to getthe parameters a(0), giving simulated measured parameters aS(1), aS(2), . . .
. Eachsimulated measured parameter set yields a point aS(i) − a(0) . Simulate enough datasets and enough derived simulated measured parameters, and you map out the desiredprobability distribution in M dimensions.In fact, the ability to do Monte Carlo simulations in this fashion has revo-Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).rimentalrealizationChapter 15.691syntheticdata set 1Mactualdata setχ2minχ2 Monte Carloparameters(s)mina1syntheticdata set 2a2syntheticdata set 3a3syntheticdata set 4a4(s)fittedparametersa0(s)(s)Figure 15.6.2. Monte Carlo simulation of an experiment. The fitted parameters from an actual experimentare used as surrogates for the true parameters. Computer-generated random numbers are used to simulatemany synthetic data sets.
Each of these is analyzed to obtain its fitted parameters. The distribution ofthese fitted parameters around the (known) surrogate true parameters is thus studied.lutionized many fields of modern experimental science. Not only is one able tocharacterize the errors of parameter estimation in a very precise way; one can alsotry out on the computer different methods of parameter estimation, or different datareduction techniques, and seek to minimize the uncertainty of the result accordingto any desired criteria. Offered the choice between mastery of a five-foot shelf ofanalytical statistics books and middling ability at performing statistical Monte Carlosimulations, we would surely choose to have the latter skill.Quick-and-Dirty Monte Carlo: The Bootstrap MethodHere is a powerful technique that can often be used when you don’t knowenough about the underlying process, or the nature of your measurement errors,to do a credible Monte Carlo simulation.
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