Degree of corroboration

Clearly a number of approximations have been used in deriving this estimate, which cannot be claimed as exact. However, a degree of corroboration has been given by Cameron s alternative statistical treatment based on applying an F-distribution to a linear system sampled at discrete time intervals (Cameron etal., 1998). Putting... 

Therefore, since language is so important, contextnal care mnst be taken to correctly understand the notions that figure in Popper s langnage. For example, mnch has been written in the recent systematics literature on a formalism that Popper nsed to characterize the degree of corroboration (C) of a scientific hypothesis, which increases with severity of test. Thns,... 

The objective interpretation of probability calculus (Popper, 1976 48, and Appendix IX, Third Comment [1958]) is necessary because no result of statistical sampling is ever inconsistent with a statistical theory unless we make them with the help of. .. rejection rules (Lakatos, 1974 179 see also Nagel, 1971 366). It is under these rejection rules that probability calculus and logical probability approach each other these are also the conditions under which Popper explored the relationship of Fisher s likelihood function to his degree of corroboration, and the conditions arise only if the random sample is large and (e) is a statistical report asserting a good fit (Farris et ah, 2001). In addition to the above, in order to maintain an objective interpretation of probability calculus, Popper also required that once the specified conditions are obtained, we must proceed to submit (e) itself to a critical test, that is, try to find observable states of affairs that falsify (e). 

The claim conld be made that systematics can proceed without underlying universal laws, for what is reqnired is nothing bnt a method that allows us to choose a preferred hypothesis from a set of competing hypotheses of relationships relative to some theory such as evolutionary theory. Indeed, we do have a method at our disposal that allows us to do just this, but is it Popperian in its logic The conformity of parsimony analysis with Popper s falsificationism has been asserted in terms of Popper s concepts of logical probability, explanatory power, degrees of corroboration and severity of test. Let us look at these concepts in more detail. 

The ties of likelihood to nondeductive inference are further revealed by the fact that likelihood searches for similarity in different situations, so that from the experience of one situation, something may be learned about another, similar situation (Edwards, 1992 199). Evidendy, this entails reasoning from past situations to future similar experiences, which cannot be compatible with Popper s notion of degree of corroboration, which says something only about the past performance of a theory but nothing about its future performance (Popper, 1976 85). 

In this, and only in this case, it will therefore be possible to accept Fisher s likelihood function as an adequate measure of degree of corroboration. We can interpret, vice versa, our measure of degree of corroboration as a generaHzation of Fisher s HkeHhood function a generalization which covers cases snch as a comparatively large 8, in which Fisher s likelihood function would become clearly inadequate. For the likeHhood of h in the light of the statistical evidence e should certainly not reach a value close to its maximum merely because (or partly because) the available statistical evidence e was lacking in precision. 

A larger 8 indicates lack of precision in e, which is accommodated by degree of corroboration (or explanatory power) but not by the likelihood function. As Popper stated, It is unsatisfactory, not to say paradoxical, that statistical evidence e, based upon a million tosses and 8 = 0.00135, may result in numerically the same likelihood. .. as would statistical evidence e, based on only a hundred tosses and 8 = 0.135. ... 

Having said all of this, it is important to remember, however (Popper, 1976 Appendix IX), ... that non-statistical theories have as a rule a form totally different from that of the h here described, that is, they are of the form of a universal proposition. The question thus becomes whether systematics, or phylogeny reconstruction, can be construed in terms of a statistical theory that satisfies the rejection criteria formulated by Popper (see footnote 1) and that, in case of favorable evidence, allows the comparison of degree of corroboration versus Fisher s likelihood function. As far as phylogenetic analysis is concerned, I found no indication in Popper s writing that history is subject to the same logic as the test of random samples of statistical data. As far as a metric for degree of corroboration relative to a nonstatistical hypothesis is concerned. Popper (1973 58-59 see also footnote 1) clarified. 

In fact, nothing can be further removed from my aims. I do not think that degrees of verisimilitude, or a measure of truth content, or falsity content (or, say, degree of corroboration, or even of logical probability) can ever be numerically determined, except in certain limiting cases (such as 0 and 1). 

How, then, is it possible to confound Fisher s likelihood function with Popper s notion of degree of corroboration Popper (1976 Appendix IX) proved to his satisfaction that the identification [of his notion] of degree of corroboration or [Carnap s notion of] confirmation with [statistical] probability (and even with likelihood) is absurd on both formal and intuitive grounds it leads to self-contradiction. ... 

Popper rejects the probabilification of science and the goal of maximal (statistical) probability, because these are part of the verificationist strategy. For Popper, the content of a theory increases with its (logical) improbability. In contrast, the degree of corroboration for a statistical theory is only of interest if the attempt is being made to refute it, that is, to reduce its probability to 0 or close to 0, because nothing is easier than to select statistical data in support of a statistical hypothesis (Popper 1976 Appendix IX). 

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