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jimrandomh comments on On Debates with Trolls - Less Wrong
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I think you're problem is you don't understand what the issues at stake are, so you don't know what you're trying to find.
You said:
But then when you found a well known book by Popper which does have those words, and which does discuss Bayes' equation, you were not satisfied. You asked for something which wasn't actually what you wanted. That is not my fault.
You also said:
But you don't seem to understand that Popper's solution to the problem of induction is the same topic. You don't know what you're looking for. It wasn't a change of topic. (Hence I thought we should discuss this. But you refused. I'm not sure how you expect to make progress when you refuse to discuss the topic the other guy thinks is crucial to continuing.)
Bayesian updating, as a method of learning in general, is induction. It's trying to derive knowledge from data. Popper's criticisms of induction, in general, apply. And his solution solves the underlying problem rendering Bayesian updating unnecessary even if it wasn't wrong. (Of course, as usual, it's right when applied narrowly to certain mathematical problems. It's wrong when extended out of that context to be used for other purposes, e.g. to try to solve the problem of induction.)
So, question: what do you think you're looking for? There is tons of stuff about probability in various Popper books including chapter 8 of LScD titled "probability". There is tons of explanation about the problem of induction, and why support doesn't work, in various Popper books. Bayesian updating is a method of positively supporting theories; Popper criticized all such methods and his criticisms apply. In what way is that not what you wanted? What do you want?
So for example I opened to a random page in that chapter and found, p 183, start of section 66, the first sentence is:
This is a criticism of the Bayesian approach as unscientific. It's not specifically about the Bayesian approach in that it applies to various non-Bayesian probabilistic approaches (whatever those may be. can you think of any other approaches besides Bayesian epistemology that you think this is targeted at? How would you do it without Bayes' theorem?). In any case it is a criticism and it applies straightforwardly to Bayesian epistemology. It's not the only criticism.
The point of this criticism is that to even begin the Bayesian updating process you need probability estimates which are created unscientifically by making them up (no, making up a "prior" which assigns all of them at once, in a way vague enough that you can't even use it in real life without "estimating" arbitrarily, doesn't mean you haven't just made them up).
EDIT: read the first 2 footnotes in section 81 of LScD, plus section 81 itself. And note that the indexer did not miss this but included it...
Probability estimates are essentially the bookkeeping which Bayesians use to keep track of which things they've falsified, and which things they've partially falsified. At the time Popper wrote that, scientists had not yet figured out the rules for using probability correctly; the stuff he was criticizing really was wrong, but it wasn't the same stuff people use today.
Is this true? Popper wrote LScD in 1934. Keynes and Ramsey wrote about using probability to handle uncertainty in the 1920s although I don't think anyone paid attention to that work for a few years. I don't know enough about their work in detail to comment on whether or not Popper is taking it into account although I certainly get the impression that he's influenced by Keynes.
According to the wikipedia page, Cox's theorem first appeared in R. T. Cox, "Probability, Frequency, and Reasonable Expectation," Am. Jour. Phys., 14, 1–13, (1946). Prior to that, I don't think probability had much in the way of philosophical foundations, although they may've gotten the technical side right. And correct use of probability for more complex things, like causal models, didn't come until much later. (And Popper was dealing with the case of science-in-general, which requires those sorts of advanced tools.)
The English version of LScD came out in 1959. It wasn't a straight translation; Popper worked on it. In my (somewhat vague) understanding he changed some stuff or at least added some footnotes (and appendices?).
Anyway Popper published plenty of stuff after 1946 including material from the LScD postscript that got split into several books, and also various books where he had the chance to say whatever he wanted. If he thought there was anything important to update he would have. And for example probability gets a lot of discussion in Popper's replies to his critics, and Bayes' theorem in particular comes up some; that's from 1974.
So for example on page 1185 of the Schilpp volume 2, Popper says he never doubted Bayes' theorem but that "it is not generally applicable to hypotheses which form an infinite set".