5 in which case you use the gedanken technique 36

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5 In which case you use the gedanken technique!
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– 36 – Suppose, however, that you are interested in an N i (for “ N interested ”) subset of the N param- eters, and you want to know their variances (and covariance matrix) without regard to the values of the other ( N N i ) parameters . You could use the gedanken technique, but you can also use the “non-gedanken easy way” by using the following procedure [see NR § 15.6 (Probability Distribution of Parameters in the Normal Case)]. 1. Decide which set of parameters you are interested in; call this number N i and denote their vector by a i . Here we use the above example and consider N i = 2 and a i = [ a 0 , a 2 ]. 2. From the N × N covariance matrix [ α χ ] - 1 , extract the rows and columns corresponding to the N i parameters and form a new N i × N i covariance matrix [ α χ ] - 1 i ; in our case the original covariance matrix is [ α χ ] - 1 = XXI = 1156 . 8125 303 . 000 18 . 4375 303 . 000 81 . 000 5 . 000 18 . 4375 5 . 000 0 . 31250 (9.2a) and it becomes [ α χ ] - 1 i = XXI = bracketleftBigg 1156 . 8125 18 . 4375 18 . 4375 0 . 31250 bracketrightBigg . (9.2b) 3. Invert this new covariance matrix to form a new curvature matrix [ α χ ] i . The elements differ from the those in the original curvature matrix. 4. As usual, we have Δ χ 2 a i = δ a T i · [ α χ ] i · δ a i , (9.3) so find the locus of a i such that the integrated probability of Δ χ a i for ν = N i contains 68 . 3% of the space; e.g. for ν = 2 this is Δ χ a i = 2 . 3. You may well wonder why, in steps 2 and 3, you need to derive a new curvature matrix from the extracted elements of the original covariance matrix. Why not just use the extracted elements of the original curvature matrix? To understand this, read NR’s discussion surrounding equation (15.6.2); this is not very clear, in my opinion. I find it easier to recall the way covariance matrices propagate errors according to the derivatives of the quantities derived, as in Cowan’s equation 1.54. I could explain this here but, quite frankly, don’t have the time; maybe in the next edition of these notes!
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– 37 – 9.6. Important comments about uncertainties Having said all the above, we offer the following important Comments : The easiest way to calculate these (hyper)surfaces is to set up a grid in N i -dimensional space of trial values for δ a i and use a contour plot or volume plot package to plot the loci of constant Δ χ 2 a i . The procedure described in § 9.5 works well for linear fits, or nonlinear fits in which the σ a are small so that Δ χ 2 is well-approximated by the second derivative curvature matrix. This is not necessarily the case; an example is shown in BR Figure 11.2. Here, the higher-order curvature terms are important and it’s better to actually redo the fit for the grid of trial values of a i as described above in § 9.2, 9.3, and 9.4. In other words, use the gedanken technique.
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