In 1983, in an open letter to the journal Nature, Karl Popper and David Miller set forth a particularly strong critical argument which sought to demonstrate the impossibility of inductive probability. Since its publication the argument has faced many criticisms and we argue in this article that they do not reach their objectives. We will first reconstruct the demonstration made by Popper and Miller in their initial article and then try to evaluate the main arguments against it. Although it is possible to conceptualize logically the idea of induction, it is shown that it is not possible on traditional Bayesian grounds.
There is a tension in Bas van Fraassen"s work between the way he often lays out the implications of accepting a theory and his agnosticism towards the unobservable entities that are postulated by our scientific theories. In the Scientific Image, for example, he claims that to accept a theory "involves as belief only that it is empirically adequate" (van Fraassen 1980, 12) and that to believe that a theory is empirically adequate is to believe "that what the theory says about what is observable (by us) is true" (van Fraassen 1980, 18). However, he also claims that what our theory says about the observables implies that there are specific unobservable entities and properties. Therefore, given that our beliefs are closed under known implications, the acceptance of a theory must also involve the belief that what the theory says about what is unobservable is also true. Now this is inconsistent.I shall call this the closure problem. Michael Friedman (1982) has introduced it in the literature and it has been controversial ever since. Here are a few contentious examples.
The p-value is the probability under the null hypothesis of obtaining an experimental result that is at least as extreme as the one that we have actually obtained. That probability plays a crucial role in frequentist statistical inferences. But if we take the word 'extreme' to mean 'improbable', then we can show that this type of inference can be very problematic. In this paper, I argue that it is a mistake to make such an interpretation. Under minimal assumptions about the alternative hypothesis, I explain why 'extreme' means 'outside the most precise predicted range of experimental outcomes for a given upper bound probability of error'. Doing so, I rebut recent formulations of recurrent criticisms against the frequentist approach in statistics and underscore the importance of random variables.
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