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534.5.1 Typical behavior: average gain and goodputWe here provide expressions for the average pruning gain as well as the goodput which jointlyconsiders gain and reliability. This is followed by several clarifying examples.In terms of the average pruning gain, it is straightforward that this takes the form G :=E v,w G(v) = (∑ ρf=1 p ) −1,fɛ f and similarly the average relative gain takes the form Ev,w r(v) =∑ ρf=1 p f(1−ɛ f ). We recall that reliability is given by ɛ 1 .In combining the above average gain measure with reliability, we consider the (average) goodput,denoted as U, which for the sake of simplicity is here offered a concise form of a weightedproduct between reliability and gain,U := ɛ γ 11 Gγ 2(4.23)for some chosen positiveγ 1 ,γ 2 that describe the importance paid to reliability and to pruning gainrespectively.We proceed with clarifying examples.Example 11 (Average gain with error uniformity) In the uniform error setting where the probabilityof erroneous categorization of subjects is assumed to be equal to ɛ for all categories , i.e.,where ɛ f = ɛ = 1−ɛ 1ρ−1, ∀f = 2,··· ,ρ, it is the case thatG = ( p 1 +ɛ−p 1 ɛρ ) −1 . (4.24)This was already illustrated in Fig. 4.2. We quickly note that forp 1 = 1/ρ, the gain remains equalto1/p 1 irrespective of ɛ and irrespective of the rest of the population statistics p f , f ≥ 2.Example 12 (Average gain with uniform error scaling) Now consider the case where the uniformerror increases with ρ as ɛ = max(ρ−β,0)ρλ, β ≥ 1. Then for any set of population statistics,(it is the case thatG(λ) = p 1 [1+(ρ−β)λ]+ ρ−β −1λ), (4.25)ρwhich approaches G(λ) = (p 1 [1+(ρ−β)λ]+λ) −1 as ρ increases. We briefly note that, asexpected, in the regime of very high reliability (λ → 0), and irrespective of {p f } ρ f=2, the pruninggain approaches 1 p 1. In the other extreme of low reliability (λ → 1), the gain approaches ɛ −11 .We proceed with an example on the average goodput.Example 13 (Average goodput with error uniformity) Under error uniformity where erroneouscategorization happens with probability ɛ, and forγ 1 = γ 2 = 1, the goodput takes the formU(ɛ) = ɛ+(1−ɛρ)ɛ+p 1 (1−ɛρ) . (4.26)To offer insight we note that the goodput starts at a maximum of U = 1 p 1for a near zero value ofɛ, and then decreases with a slope ofδUδɛ =p 1 −1[ɛ+p 1 (1−ρɛ)] 2, (4.27)which as expected 4 δis negative for all p 1 < 1. We here see thatδp 1δ U δɛ | ɛ→0 → 2−p 1which isp 3 1positive and decreasing inp 1 . Within the context of the example, the intuition that we can draw isthat, for the same increase inɛ 5 , a search for a rare looking subject (p 1 small) can be much moresensitive, in terms of goodput, to outside perturbations (fluctuations in ɛ) than searches for morecommon looking individuals (p 1 large).4. We note that asking for|S| ≥ 1, implies thatɛ+p 1(1−ρɛ) > 1 δU(see (4.24)) which guarantees that is finite.n δɛ5. An example of such a deterioration that causes an increase in ɛ, can be a reduction in the luminosity around thesubjects.