Talk:Probability interpretations

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[edit] Missing interpretations

A number of important interpretations are missing. The propensity interpretation of Popper as well as the various logical interpretations of probability. Then we have the formalist view. This page should contain 'all' views I think. INic 22:35, 18 April 2006 (UTC)

I think it's safe to say that the propensity interpretation of Popper has not been adopted in practice to any significant extent. I'm not sure what you mean by "the various logical interpretations" or the "formalist view" - you haven't given any links or explanation, but I'm confident that the same wll be true. Feel free to expand these in probability interpretations, but I think it's clear that in practice there are two main views (frequentism and Bayesianism) each of which is dominant in some professional groups and a minority in othersJQ 09:35, 5 May 2006 (UTC).

I agree with INic. The main Probability article says more about probability interpretations than this supposedly more detailed article does! This is currently a fairly poor article. Ben Finn 15:55, 31 December 2006 (UTC)

Ditto. The "frequency interpretation" is really only of historical interest, within philosophy at least, since I know of no present philosopher who advocates it. (The well-known frequentists are Venn, Reichenbach and possibly von Mises.) I wonder then if John Quiggin is referring to what we philosophers call frequency theories of propensity? (E.g. the views of (possibly) von Mises, Popper, Miller, and Gillies). While these face substantial difficulties, there are at least contemporary adherents. I have also heard physicists refer to objective propensities as "frequency probabilities", on the grounds that they're estimated empirically through measuring relative frequencies. But this terminology is rather misleading in my view.

Also, concerning degree-of-belief type probabilities, there are indeed a range of views not covered by the article. The notion of epistemic probability is based on the idea that degrees of belief are subject to rational constraints, so that there are "correct" and "incorrect" degrees of belief in a given state of knowledge. An extreme case of this is the logical interpretation, where degrees of belief are fixed by logic alone. Bayesianism isn't an interpretation of probability, but a theory of confirmation. It's true that Bayesians assume some sort of subjectivist interpretation of probability, but the exact form of this varies quite a bit from one Bayesian to another. Some, so-called "Objective Bayesians", use an epistemic interpretation.--137.82.40.29 21:18, 22 March 2007 (UTC)Richard Johns

Richard I agree completely, just a few remarks. We should mention that there are different kinds of Bayesianism here and explain the differences but not go too deep into it as Bayesianism has it's own page devoted to that. When it comes to "frequentism" I agree that the detailed accounts by Reichenbach and von Mises for example only are of historical interest today. But "frequentism" in a more general sense do have a lot of followers today, as it's the default interpretation taught in all ordinary university courses in probability theory. It's the by far most common opinion among working statisticians, probability theorists and physicists for example. When Bayesians object to "frequentist" reasoning, for example, they object to this dominating view which is in general taught today. iNic 00:45, 23 March 2007 (UTC)

It's not taught in all probability courses, but it is taught in (almost?) all basic statistics courses. Michael Hardy 01:38, 23 March 2007 (UTC)

Michael and INic: Your statement that frequentism is taught in stats courses is puzzling to me. Do you mean that frequentist methods of statistical inference are taught in those classes? Of course that's true, but frequentism as a method of stat. inference (all those p-values, confidence intervals, null hypotheses, etc.) is quite different from frequentism as an interpretation of probability. Actually, after posting my comments yesterday I wondered whether Quiggin was using "frequentism" to refer to frequentist statistical methods. That would explain why he regards frequentism as a direct competitor to Bayesianism. It is all rather confusing, as R.A. Fisher (the founder of frequentist stats) was strongly drawn to the ideal of objectivity in stats, and so (I would guess) used some sort of frequentist interpretation of probability. But in this article we need to distinguish clearly between interpretations of probability and theories of statistical inference/ confirmation. They are quite different projects.--64.180.160.210 17:52, 23 March 2007 (UTC)Richard Johns

I'm not sure that we can say that the projects are that different, really. I would say that they are just the opposite sides of the same coin; one philosophical/ontological and one practical/methodological. And the dependence between the projects is even closer in the "frequentist" case because here the current philosophical definition is via the statistical methods used (and hence, here we get some different philosophical sub-schools due to the existence of some competing statistical methods). In the Bayesian camp they also say that the connection between philosophy and practicality is very close, as they claim that their statistical methods can be derived from their respective ontological core theories. But nevertheless, I agree with you that this article should stress the philosophical side of the matter. iNic 01:43, 25 March 2007 (UTC)

INic: I'm glad we agree that this article ought to focus on the meanings of probability rather than on theories of statistical inference that might be associated with such meanings. That's all that matters here, I think. I think I'll go ahead and make some changes to refocus the article on interpretations of probability.--Richardajohns 05:21, 3 April 2007 (UTC)

Your introduction is good; we need a general introduction like that. However, the traditional classification of interpretations into subjective and objective is a little bit misleading I think. Not all Bayesian interpretations are subjective. In fact, most of them try really hard to get rid of the subjective label by introducing a theory only applicable to rational men for example. Some even claim that Bayesian probability is more objective than reality itself, as it ought to be viewed as a natural extensions to logic. And current frequentist interpretations have been criticized for not being absolutely objective; the statistical methods and models used at a particular instance are ultimately due to the personal judgement of the statistician herself. Bottom line is that I don't know if modern frequentists ever claimed that they are absolutely objective, nor do I think that most Bayesians claim that their theory is absolutely subjective. Therefore, I think we obtain a better characterization of the two groups of interpretations if we instead stress that in one of the groups "probability" is always tied to a conceptual experiment, while in the other group "probability" is always tied to the concept of a statement in a language. iNic 14:24, 4 April 2007 (UTC)

Well, you're right that the terms "objective" and "subjective" aren't without difficulty. As you point out, the subjective interpretations include epistemic and even logical probability, which aren't subjective in the sense that they are subject to rational standards. But they are still subjective in the sense that they depend on the belief or knowledge of a thinking subject, albeit an idealised one. Moreover, tying probability to a statement in a language doesn't make it subjective in this sense. After all, physical outcomes of experiments can also be expressed in statements. "Personal" is another option, but it's probably worse that "subjective" for seeming to be beyond the realm of rationality.

My intention was to include other logical interpretations too, not only Bayesian ones, into the "language statements" group. You know, the logical-language interpretation by Carnap for example. This is the main problem with the subjective/objective distinction I think; on the surface it seems to be an exhaustive classification, but all we really mean with "subjective" is Bayesianism of different kinds (even those with pure objective claims), and by "objective" all we mean is frequentism of different kinds (even though they are all somewhat subjective). So what at first sight seems to be a nice all-inclusive classification turns out to be contradictory, narrow end excluding. Carnaps theory, for example, neither fit into the "objective" nor the "subjective" camp using this terminology. iNic 02:31, 7 April 2007 (UTC)

Perhaps it should be changed to something like: "Subjective probability, on the other hand, can be assigned to any statement whatsoever, even when no random process is involved, as a way to represent its subjective plausibility, degree of support by the available evidence, or rational degree of belief". (?)

Yes this is better. But when the 'belief' is assigned only to robots for example, should we still talk about "subjective" probability? It all gets very confusing when trying to makes sense of this distinction, I think. iNic 02:31, 7 April 2007 (UTC)

As for "objective", I don't think we want to replace it with "experimental", as objective probabilities apply to events outside the lab, beyond human control. (E.g. the probability that a 40-year-old Canadian will die in the next year.) I guess you're right in saying that frequencies aren't necessarily all that objective, as there is always the problem of choosing a suitable reference class, but the idea in the introduction is just to give the basic outline, and worry about the details later.

Conceptual experiments isn't confined to human control nor within the walls of a lab. If the word 'experiment' is misleading to you and others we can use 'sample space' or 'reference class' to convey the same basic idea here. The problem with the reference class that you refer to illustrates this very well. For a frequentist that isn't a problem at all, as frequentists doesn't speak about probabilities unless a reference class (=experiment=sample space) is defined. What is left when a reference class isn't defined is only a statement about something. This statement can be interesting to Bayesians of various kinds to consider, as well as to logical language constructors such as Carnap, but not to frequentists. iNic 02:31, 7 April 2007 (UTC)

To use your example; to frequentists there is a probability attached to the process of randomly picking a 40-year old Canadian out of all (or some pre-defined set of) 40-year old Canadians and see if (s)he will die the next year. If this (or some other) process (=experiment) isn't defined, well then there simply is no frequency probability defined. For example, if you meet someone at a party and (s)he turns out to be a 40-year old Canadian, we can't say anything about any probability that (s)he will die the next year, if we are frequentists. The probability is completely undefined. The reason is that no random experiment is defined here. And no random experiment means no sample space. And no sample space means no probability space. And no probability space means no probability in the sense of Kolmogorov. iNic 02:31, 7 April 2007 (UTC)

The formulation I came up with, that objective probabilities "reveal themselves when a type of event occurs at a persistent rate, or relative frequency, in a long run of trials" is based on a paper by Jerzy Neyman "Frequentist Probability and Frequentist Statistics", Synthese 36 (1977) 97-131. He stresses that the frequentist concept of probability is founded upon the apparent stability of relative frequencies. Frequentists and propensity theorists of all stripes should be happy with that, I think.--Richardajohns 01:35, 5 April 2007 (UTC)

Yes, this is a good first approximation to what frequentism means. And propensity theorists might find it cool too. But are these two the only "objective" probability interpretations? I would say that Carnap's theory, for example, is objective too. But Carnap wouldn't be happy with Neyman's characterization of his theory... This obj/subj-jargon is very confusing to the lay person I think. Unfortunately, we do have to mention this because it is traditionally used, but at the same time I think we have to warn the reader that these common words are used in a very limited and odd way in this context. iNic 02:31, 7 April 2007 (UTC)

Ok iNic, I think you're right about both "objective" and "subjective" being confusing. I've made changes that I hope are headed in the right direction, at least. Feel free to make further modifications. Briefly, I replaced "objective" with "physical", and explained it in a bit more detail. I replaced "subjective" with "evidential", which I hope is general enough. I also listed the different kinds of physical and evidential probability that (I think) should be covered by the article. Tell me if you agree. The terminology now fits pretty well with that of the article "Bayesian Probability" on Wikipedia, which is a plus.Richardajohns 21:14, 8 April 2007 (UTC)

Now I've added a short section on logical, epistemic and inductive probability. I'm not sure these can be dealt with separately, as there's so much overlap between them. There could be individual subsections on Keynes, Carnap, Ramsey, etc. by someone knowledgeable about them (not me).Richardajohns 22:34, 10 April 2007 (UTC)

I applaud your good work! :-) This article is rapidly approaching a very good final state, due to your efforts. It would be nice to have some pictures attached to the new sections too. Would it be wrong to have a picture of Popper as an illustration to the propensity interpretation (as the connection Popper-propensity is rather strong)? The logical interpretation I don't know what image to use. Do you mind if I take some of your text from the Propensity interpretation section to start off its proposed main article? Sub-sections about Keynes, Carnap and Ramsey would be nice to have at that main article I think, so we can keep the sections here rather short. iNic 01:51, 11 April 2007 (UTC)

I'm glad we're finding some consensus at last! Thank you for your criticisms, which have definitely improved the article. I agree that pictures would be nice, and that Popper is the most suitable choice for the propensity interpretation, even though C. S. Peirce seems to have thought of it first. For the logical interpretation Carnap is the main figure, but Keynes came earlier, so perhaps have pictures of both? I see that you plan to make a separate article on propensity, which is a good idea. Feel free to split the existing text as you see fit -- I plan to add quite a bit more to the propensity article when I get some time. (It might get tricky when it comes to presenting my own theory though!) Richardajohns 17:18, 11 April 2007 (UTC)