88 Andrew C. Lorenc
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Fig.5.6 As Fig. 5.4 for -log(probabilities).
Fig. 5.7 shows the expected benefit ca1culated using (35), for different values of
C. C=O corresponds to a delta function benefit; the curve is identical to the posterior pdf (similar to those shown in Fig. 5.5; this example is for a more ac curate but
equaUy unreliable observation). The analysis value which gives the greatest
expected benefit is shown by x, and corresponds to the maximum of the posterior
pdf. In practice, an analysis which is as accurate as the background x b is stiU of
some use. So it is more plausible that the width of the benefit function should be
similar to that of the background pdf (which has variance 9 in our example). The
expected benefit curve for this case has its maximum at +. It has chosen the peak
from the posterior pdfwhich has the largest area (0.61 compared to 0.39). FinaUy,
if we expand the width of the benefit function so that it is large compared to the
separation between the peaks, then we get the curve with maximum at D. For very
large C the Gaussian benefit function becomes a quadratic, and the maximum is
always at the mean value of the posterior pdf.