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One point says: "For the frequentist a hypothesis is a proposition (which must be either true or false), so that the frequentist probability of a hypothesis is either one or zero. In Bayesian statistics, a probability can be assigned to a hypothesis that can differ from 0 or 1 if the truth value is uncertain." However, also the Bayesian approach is inherently aristotelian, i.e. an hypothesis is necessarily true or false, but we have a distribution over these truth values. This point probably mixes up Bayesianism with fuzzy concepts. Furthermore, in frequentist statistics we have a p-value, which is an error-probability of the hypothesis under test, so it generally does not assign 0/1-probabilities to hypotheses. User:193.170.104.110
measure-theoretic based probability theory, which is the formal basis of both Frequentism and Bayesianism. Measure theory is not the formal basis of Bayesianism. A weaker version of Kolmogorov's axioms is derived from the basic assumptions; that's it. The two approaches are very distinct. Dixit Jaynes (Appendix A of Proba Theory, the Logic of Science):
[The] Kolmogorov approach to probability theory [...] could hardly be more different from ours in general viewpoint and motivation; yet the final results are identical in several respects.. — Gamall Wednesday Ida (t · c) 17:54, 9 February 2017 (UTC)
Please forgive, on one hand, and weigh, on the other, my limited but perhaps more formal (than that of those now encountering Bayes’s approach for the first time) instruction on this relatively subjective reasoning tool. Full disclosure: my imperfect-but-hardly-gonzo relevant background was as a BA physics graduate at one of the premier midwestern lib-arts undergrad institutions. Technically I was a just a physics major, but the math I did take was substantial, and (IIRC correctly) I chose to take even more of it than ‘’either’’ physics or maybe even math majors had to. The course (perhaps just Introduction to ...) “Probability and Statistics” probably was a physics-major requirement, was taught by a math prof, and included a intro to Bayes’s work. Our article may, or not, cover it as well as what my recollection of my course does, but I came away with a crucial recollection perhaps worth repeat’’’shar‘’’ing, now that Bayesian statistics is ’’’at least‘’’ a candidate for epidemiology: formal math is something you can reduce to cold logic, including even formal statistics. But I was taught that Bayesian statistics uses rigorous math, and applies it to gaining ‘’mathematically’’ rigorous results, whose validity, usefulness, and value is based on its likely usefulness’’’in‘’’ making important decisions that are outside the realms where mathematical logic can *prove* anything of the kind that we usually hope for, in using ‘’’when we use’’’ it. My perhaps gonzo picture is that Bayesian methods are a more powerful tool than flipping a coin, when you’ve admitted you most likely have to make up your mind what to do, and your educated ’’best guesses’’ about the “known unknowns” seem at least mildly likely to be worth being “weighed in’’’to that decision’’’”. We were offered only a passing ‘’’description’’’, but that educated introduction to the subject, in that half-ancient ‘’’but educated’’’ introduction may be valid in supporting ‘’’my’’’ confidence that the experts are doing ‘’what is’’, in the face of the inacessibility of detailed knowledge, the next best thing to our unachievable wish.
—JerzyA (talk) 21:09 ‘’’& :50’’’, 21 February 2020 (UTC)
Metaprobability redirects here, but isn't explained in the article. -- Beland (talk) 14:56, 1 May 2020 (UTC)
The redirect Degree of belief has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 June 4 § Degree of belief until a consensus is reached. Hildeoc (talk) 00:44, 4 June 2023 (UTC)
I have difficulty understanding the section "Dutch book approach". A good first step may be to explain why it is sometimes used in support of bayesianism, before explaining that some non-bayesian approaches avoid Dutch Book as well. The given quote from Ian Hocking is also a bit hard to understand, since it introduces new terms such as "dynamic assumption" and "personalist". Alenoach (talk) 10:52, 14 November 2023 (UTC)
The statement, ‘Unfortunately, it is not clear how to assess the relative "objectivity" of the priors proposed under these methods’ is both unsupported by any references and hyperbolic. It is certainly possible to assess whether a prior respects a symmetry such as a transformation group (e.g., translation invariance), one of the methods specifically listed. Whether this is “unfortunate” or not, it is at least clear. I have weakened the hyperbole slightly. Jmacwiki (talk) 03:32, 25 March 2024 (UTC)