Talk:Classical definition of probability
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[edit] Classical probability not a special case of Bayesian probability
The last sentencve of the introduction to this article claims:
“The classical definition enjoyed a revival of sorts due to the general interest in Bayesian probability, in which the classical definition is seen as a special case of the more general Bayesian definition.” [My italics]
The italicised last clause of this sentence should be removed as logically confusing. For certainly as Bayesian probability is defined in the Wikipedia article referred to here, it is not a logically special case of the Bayesian definition, but a logically independent and mutually exclusive definition, as immediately evident from placing the two definitions side by side as follows:
Classical definition: “The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible.”
Bayesian probability is an interpretation of the probability calculus which holds that the concept of probability can be defined as the degree to which a person (or community) believes that a proposition is true.”
If there are some people who see the first definition as a special case of the second, their logical confusion should not be reproduced in this article. However, since no source is given to demonstrate anybody who accepts these definitions of classical and of Bayesian probability believes the first is a special case of the second., I propose deletion at least until such as time as evidence of such is provided and can be discussed.
It also remains to provide sourced evidence that this classical definition as defined here did indeed enjoy a revival due to general interest in Bayesian probability as defined here.--Logicus 14:39, 14 October 2007 (UTC)
- You keep showing your misunderstanding of Bayesian probability theory. These two definitions seem different because they both deemphasize their root which is Bayes theorem. The important thing to recognize about Bayesian probability is that it is not just a degree of belief, its a degree of belief that follows Bayes theorem. Hence the name: "Bayesian". The Classical definition is a specific application of Bayes theorem where prior beliefs are defined as: "nothing leads us to expect that any one of these cases should occur more than any other" . Modern Bayesian probability theory generalizes this definition by allowing different priors according to the situation. You need to look further in Laplace's book to see that he is pioneering the Bayesian methods. Throughout the book the Bayesian optimal-degree-of-belief interpretation is clearly and consistently used. I don't know if the revival of the classical definition is due to this, but when you look at the equations it is pretty clear that Laplace was using Bayesian methods. --BenE 21:58, 14 October 2007 (UTC)
Although I disagree strongly with almost everything that BenE says here (and elsewhere) I think it is a good and valid point that Bayesianism is a movement that want to return to old thinking (but with a new twist). ("Intelligent Design" comes to my mind as a similar case...) Therefore I think it's both correct and appropriate to say here that the Bayesians are trying to bring back the classical definition to life but in a new disguise. This is why Bayesians think that the classical definition still makes sense. However, the opposite is of course not true; that Laplace (or anyone else at that time) were a Bayesian. That is just silly. iNic 00:40, 15 October 2007 (UTC)
Logicus's to BenE & iNic: I'm in a rush here, but at a glance it seems yet again we have a literacy problem here. Your criticisms are not relevant because my valid point is a logical and pedagogical one about the current definitions in thse article, whereas you are contesting these definitions. My edit should be restored until alternative logically and historically valid conceptions have been worked out and sourced.--Logicus 18:15, 15 October 2007 (UTC)
The very simple point here is that as the definitions are currently actually stated in these two articles, the classical definition is clearly not a logical special case of the Bayesian, whatever the historical rights and wrongs of this might be, and so the claim made that some people see it as such is just confusing. To show this claim is true it must be demonstrated that people who accept the characterisation of the classical definition GIVEN HERE as correct see it as a special case of the Bayesian probability definition as stated in the other article. I suspect there are no such people, but rather there have been RE-DEFINITIONS, or as iNic says, "a new disguise". I also suspect the classical definition has not enjoyed a revival, but rather has been redefined somehow and called ‘the classical definition’. When we are doing history of ideas it is vitally important to provide historically accurate linguistic characterisations in order to capture the dialectic of the continual criticism and revision of ideas and their problemshifts and concept stretchings etc. Thus I restore my edit.--Logicus 18:11, 17 October 2007 (UTC)
Logicus to BenE: Given your expertise on Laplace and since you apparently subscribe to probabilist philosophy of science and so maintain all scientific reasoning is probabilist, would you be so kind as to explain how the classical definition of probability you claim/accept Laplace subscribed to was applied by him to Newton's law of gravity in coming to his strong belief that it was true? What probability did Laplace give that law ? And especially what probability did he give it in 1787, when he reportedly demonstrated it could give a better explanation of the secular acceleration of the Moon's apogee than Euler's alternative law and Cartesian explanation that had won the 1770 Paris Academy prize for explaining it ?--Logicus 18:11, 17 October 2007 (UTC)
BenE's response: I don't know, but this question goes beyond probability theory. I'm not sure that Laplace saw his definition as a definition for a Philosophy of science. It might have been confined to describing a probability theory. One could argue that's one of the reson it's a special case of today's bayesianism which, for a lot of people, is also a philosophy of science. I continue to disagree that the definitions are different. Laplace simply described in his own words (Laplace has a habbit of describing equations in words) a special case of what the word Bayesian represents: Bayes theorem. Il'll restore the article since the majority is against Logicus' changes.--BenE 19:43, 17 October 2007 (UTC)
- Logicus, the definitions in an encyclopedia doesn't have to be logically derivable from one another in the way you indicate. As you point out, the historical context is more important to have in mind. This is exactly why it's so silly to call Laplace or Bayes Bayesians. However, the current Bayesian movement often claims that Laplace really used modern Bayesian methods. In this way it's a fact that Bayesians today think that Laplace really was the first Bayesian. It's a part of their self-image and a way to justify their philosophy, I guess. It's of course not true, but it's true that they believe it's true. Anyway, for this reason some (but for sure not all) of the previously abandoned reasonings of Laplace are viewed as correct by Bayesians today. In this way Laplace's general philosophical outlook suddenly has gotten modern adherents. iNic 23:36, 17 October 2007 (UTC)
- However, this article is far from complete. I will add a section about the paradoxes the classical definition led to as well as a short account of the major critics of the classical definition, Venn and Boole, and how they transformed the subject to become a more scientific discipline. When that is added it will be easier (I hope) to follow the history of ideas leading to frequentism and eventually Bayesianism. iNic 23:36, 17 October 2007 (UTC)
The confusion of Jefferys & BenE on Bayesian probability
From the 19 Sept and 13 Oct contribution of Jefferys and those of 20 Sept and 10 Oct to the Bayesian probability article discussion page it seems that the conception of probability they have in mind is what some authors call 'logical' or 'objective Bayesian' probability as distinct from 'subjective probability' or 'subjective Bayesian probability' which that article identifies with 'Bayesian probability' in its opening sentence. It is the latter that Logicus has been employing in his analyses, but suggested might be a category mistake in his constructive analysis of 3 October. This would explain why Jefferys thinks that if there is only one hypothesis about some domain, it must have probability 1, even though not a tautology, because in this conception of probability, when there are n hypotheses it sets priors for each as 1/n. Thus it seems the gross unfamiliarity of Jefferys and BenE with the literature on Bayesian probability has caused much timewasting rather than getting on with the constructuve busines of conceptual clarification of definitions of probability this article requires. Instead it has led to them lecturing and haranging Logicus--Logicus 18:18, 18 October 2007 (UTC)
Logicus to iNic
- Your proposal to add a section about the paradoxes of classical definition and Venn and Boole is a good one. It would also be good if Wikipedia could aim at providing a conceptual clear and historically adequate dialectical account of the evolution of concepts of probability, on which the literature is appallingly confusing.
- Your other remarks about are quite wrong. I never said two definitions have to be logically derivable from one another. But if an encyclopedia presents two definitions and then says some people regard one as a special case of the other, when logically and semantically it is clearly not a special case but they are mutually exclusive, then it at least needs to be explained what different definition such people might possibly be using. The importance of historical context means that historically adequate definitions must be provided, and I reckon that at least the classical definition given here is historically adequate, but suspect the definition of Bayesian probability as purely subjective probability may not be, and may need revision. It is clear from the apparently BenE's and Jefferys's remarks that like Jaynes they reject subjective probability, but claim to be Bayesians, like Jaynes. The leading question here is whether Bayesianism can be said to be anything more specific than just using Bayes's Theorem, as such as BenE wants to define it, but which all versions of 'Kolmogorov' probability do, whereby all probability is Bayesian. I shall restore my edits until such time as article has adequately redefined the concepts.--Logicus 16:14, 19 October 2007 (UTC)
Logicus to iNic: Note the classical definition is notoriously essentially circular, defining probability in terms of (equal) probability. Something needs to be said about this fatal defect, breaching the classic Aristotelian requirement of adequate definition. And where are the problems of principle of indifference ? Forthcoming ? --Logicus 14:51, 20 October 2007 (UTC)
