Fisher’s likelihood function

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). 

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. 

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. 

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. ... 

Inference based on the likelihood function using Fisher s ideas is essentially constructive. That means algorithms can be found to construct the solutions. Efron (1986) refers to the MLE as the "original jackknife" because it is a tool that can easily be adapted to many situations. The maximum likelihood estimator is invariant under a one-to-one reparameterization. Maximum likelihood estimators are compatible with the likelihood principle. Frequentist inference based on the likelihood function has some similarities with Bayesian inference as well as some differences. These similarities and differences will be explored in Section 3.3. 

An alternate argument for minimizing S 6) is to maximize the function i( 6,a Y) given in Eq. (6.1-10). This maximum likelihood approach, advocated by Fisher (1925). gives the same point estimate 6 as does the posterior density function in Eq. (6.1-13). The posterior density function is essential, however, for calculating posterior probabilities for regions of 6 and for rival models, as we do in later sections of this chapter. 

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