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- W2017174810 abstract "An inductive method CA in the A-system of Carnap [1] is immodest, on evidence e, iff its estimate, on e, of its own accuracy is higher than its estimate, on e, of the accuracy of any rival method CA,. Immodesty seems to be a condition of stable trust: if you trusted a modest CA, you should start by trusting its advice to replace it by a rival that it estimates to be more accurate. One might guess that any CA would be immodest on any evidence. But in [2] I proved that, under a certain accuracy measure taken from Carnap [1], ?? 20-21, there would be exactly one immodest CA. Unfortunately, that one sometimes turns out to be Co (the straight rule); and since nobody in his right mind would trust Co we are then left with no acceptable CA. Stephen Spielman [3] has proposed a remedy: an estimate of accuracy, on evidence e, should disregard accuracy under circumstances that are ruled out by e. Spielman proves that if this change is made, any CA is immodest on any evidence. But Spielman forbears to mention that his remedy is exactly the one that I considered in the next-to-last paragraph of [2], and there rejected for a reason that turns out to be a bad one. So it is up to me to explain why it is that Spielman's remedy does succeed, despite what I said. What if e is null evidence: a sample of size zero ? Then e does not rule out any circumstance whatever, so Spielman's remedy makes no difference. But the case of null evidence is one of the cases in which I had proved that Co alone is immodest. So I wrote of the remedy that Spielman has now endorsed: This change might be appropriate on other grounds, but it will not solve our difficulty: Co is still uniquely immodest [on null evidence] ([2], p. 63). Since the remedy makes no difference in the case of null evidence, Spielman's results and mine seem to be in flat contradiction, casting doubt on both. He has proved that every Cx is immodest on null evidence; I proved that none but Co are. But the discrepancy is explained away if we keep track of the different approximations used by Spielman and myself. Approximations are not innocent here-not even if they become as good as you please in a large enough universe. If Cx estimates CA, to be the most accurate method, then CA is immodest iff A' = A exactly. An approximation that turns this exact equality into an approximate equality for all but one CA will radically distort the outcome. My approximations cause just such distortion, in the case of null evidence. Spielman's do not. Carnap proposes in [1], ?? 20-21, that the (in)accuracy of a method may be measured by its mean square error in estimating relative frequencies of Q-properties on the basis of samples of a fixed (arbitrarily chosen) size. Call this accuracymeasure A*. But since A* itself is mathematically intractable, Carnap derives a * Received February, 1973." @default.
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- W2017174810 date "1974-03-01" @default.
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- W2017174810 title "Spielman and Lewis on Inductive Immodesty" @default.
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- W2017174810 doi "https://doi.org/10.1086/288571" @default.
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