Talk:Imprecise probability

Rewrite section on motivation?
It seems time to clean up this page a bit. The motivating example is controversial, and uncited (it is certainly not from his 1991 book). I vote to remove the whole section completely and to start all over again, possibly using a simpler and less controversial example. Section 1.1.4 of Walley's book (page 4) would be IMO a good starting point. Any other opinions out there? Mtroffaes (talk) 10:34, 2 June 2008 (UTC)


 * I agree with Mtroffaes, this whole section isn't a good example. I am still trying to understand what imprecise probability is, but as it is this article doesn't provide much help.... Aurelein (talk) 13:30, 14 October 2008 (UTC)


 * Judging by what's written in this article, “imprecise probability” is quite a simple concept: we have a Bernoulli random variable X (tossing thumb-tack) which has certain probability p of landing pin-up (1) and 1–p for pin-down (0). Now the observer doesn't know the value of p, so he uses the fact that EX=p to estimate this quantity. He simply takes sample average and then by LLN+CLT it will be consistent and asy.normal. This estimated quantity $$\hat{p}$$ they dub "imprecise probability", although it's just simply a random variable with mean p and approximately normal distribution.


 * Why do they do that? Hell if i know... Maybe lack of knowledge of basic probability theory. Maybe cover-up for money laundering or something, seeing as they have a "Society for promotion of imprecise probability". Maybe I just didn't understand what's this all about. // Stpasha (talk) 20:05, 1 July 2009 (UTC)


 * If Experimenter A was truly an adept of the money laundering Imprecise Probability society, he would have modeled his belief by using a vacuous-prevision. In this particular case, this would mean that the experimenter says that the probability of the thumbtack landing pin-up is in the interval [0,1], representing his total ignorance. The fact that the expert in basic probability summarizes the theory so wrongly, is nothing but a pledge for a start-over of the section. Fhermans (talk) 15:36, 25 August 2009 (UTC)


 * Can we achieve a consensus to delete the motivating example, on the grounds that it has so far only caused confusion with non-experts, and hence proven to be unhelpful to understand imprecise probability? Illywhacker, I'd appreciate your approval since you have been writing part of that section, and you have showed how unhelpful the example is in the first place. (I'll try to improve other parts of the article in the mean time.) Mtroffaes (talk) 09:43, 23 September 2009 (UTC)


 * I agree with Mtroffaes (talk) about the deletion. Criticisms of imprecise probability theory commonly found in the literature can and perhaps even should be put in a separate section. Equaeghe (talk) 14:49, 23 September 2009 (UTC)


 * So as to avoid confusion about what is being suggested: I believe Mtroffaes is suggesting deleting the whole of section 3, 'Motivation...', not only section 3.1 'Alternative explanation...'. I have no problem with this plan. I do have a problem with deleting only section 3.1 'Alternative explanation...', since this leaves the motivating example unchallenged as 'encyclopedic truth'. Since this example is very misleading, implying as it does that the issue cannot be dealt with via probability theory, this is unacceptable. illywhacker; (talk) 09:36, 23 April 2010 (UTC)


 * I think there is now sufficient consensus to remove section 3. I will do that shortly. After that we should wait a bit before adding new material about motivation to see whether somebody who hasn't commented yet disagrees and would like to restore that part (please comment here first in that case). I'm also going to remove the top-of-the-page boxes, they have become moot since the end-of-last-year's edits. Equaeghe (talk) 14:31, 1 July 2010 (UTC)

Justification for undo of removal of subsection "Alternative explanation without imprecise probabilities"
The reasons given for the removal of this subsection were: 1) it is incorrect; 2) "It is not part of the book"; 3) it is original research. (1) No indication or explanation of any error is given. (2) Does Wikipedia consist of advertisements for books? (3) It is trivial high school or undergraduate material in probability theory. illywhacker; (talk) 09:32, 14 September 2009 (UTC)


 * The reasons stated strongly argue in favour of deletion of the "Motivating example" entirely, and not just the alternative explanation! Let me address the concerns again, in detail. 1) Fhermans already hinted at the error quite accurately. A typical imprecise probability approach uses a set of probabilities compatible with the available information. For instance, the Frechet bounds are a fine example where you use the set of all probabilities that are compatible with marginals rather than arbitrarily assuming independence. 2) The "Motivating example" states "Walley considers the exercise of determining the probability that a tossed thumbtack lands pin-up. Three experimenters perform this exercise, as follows (adapted from Walley)" - however there is no precise citation to Walley at all - so the reason for deletion is indeed that the citation is plainly wrong, and moreover it is not part of any work on imprecise probability that anyone knows of (and definitely not of Walley's book, which I have read). 3) Logically, if the motivating example cannot be found in the literature, then it qualifies as original research on imprecise probability theory. Having said all this, I really would like to see a more serious discussion on imprecise probability theory in this article, including a historical overview to show that there is nothing new at all about this theory and that it is as old as probability theory itself (as evidenced in for instance George Boole's "An investigation on the laws of thought" from 1854, which contains *many* motivating examples). Mtroffaes (talk) 10:51, 22 September 2009 (UTC)


 * IIRC, a very similar argument to the "Alternative explanation" is made by Edwin Thompson Jaynes in his book "Probability Theory: The Logic of Science" (chapter 18, 'The Ap-Distribution and Rule of Succession', section "Confirmation and Weight of Evidence"), as a motivation for assigning "probabilities of probablilities". If Jaynes is indeed making the same argument, then that would suffice as a reference.  --Classicalecon (talk) 11:15, 23 September 2009 (UTC)


 * Imprecise probability theory is not about probabilities of probabilities, but about sets of probabilities. This is an important point to make and the current motivational example does not allow this. There are far better, attributable examples that also allow mention of criticism (which the alternative explanation is, in essence) commonly found in the literature. I know Edwin Thompson Jaynes has formulated criticism in his posthumously published book (easily rebuked IMHO), so that could be a source. Equaeghe (talk) 15:03, 23 September 2009 (UTC)


 * But sets are simply a special case of probability distributions, so probabilities of probabilities subsumes sets of probabilities! illywhacker; (talk) 09:21, 23 April 2010 (UTC)


 * I think I must misunderstand what you mean. How or in what way can you say that a function (probability distribution) is a set? Moreover, where in the probability theory literature (precise or imprecise) is this idea proposed/used? Equaeghe (talk) 14:26, 1 July 2010 (UTC)


 * I did not say a probability distribution was a set. I said a set was a special case of a probability distribution, one that takes on a constant value on the set and zero elsewhere. Thus, the set of probabilties [p, q] ⊂ [0, 1] is represented by the uniform distribution on [p, q]. illywhacker; (talk) 13:20, 1 October 2010 (UTC)


 * I understand where you are coming from, and this is indeed a common misunderstanding to newcomers of imprecise probability theory. If you read the literature, then (aside perhaps from some obscure research) the understanding is generally that imprecise probabilities are sets of probability measures, without any further hierarchical modelling. If you put a distribution on the set, you have a classical 2nd order probability model. You don't need any of the imprecise probability machinery to deal with this. Is it now clear? Mtroffaes (talk) 17:14, 16 March 2011 (UTC)


 * I do not as yet see the difference. A set can be represented by a probability distribution. After that, any difference can only be a question of the algebraic structure used. After all, a parametric model is a set of probability distributions. illywhacker; (talk) 14:20, 29 March 2011 (UTC)

I think there is a need to distinguish Dempster-Shafer type theories and imprecise probability theory. I think the alternative explanation without imprecise probability theory might be used as a critique of Dempster-Shafer theory, but it is not relevant to imprecise probability theory in general. I think Lindley also criticises this theory.

In Appendix A of Jaynes book he gives some qualified support to the use of imprecise probability under the heading "Holdouts Against Universal Comparability".

I would argue for deletion of the motiviating example (or moving to a page on Dempster Shafer theory). I am doing this edit annonymously, but I am David Rohde and I made the deletion originally. Also I applaud the effort of whoever added all those citations to the earlier section. I must say I have reservations about having so many different topics covered by this article though...


 * Thanks. I added those citations simply to adhere to Wikipedia standard. Imprecise probability is nowadays a rather big topic (at least compared to what it was 30 years ago), so I think that this article can only be a summary of what is out there, with pointers to other articles with further details about particular theories. Dempster-Shafer theory is (at least as many of my colleagues see it) a special case within imprecise probability theory&mdash;I think this is clear from the current article&mdash;if not, feel free to improve it. However, it is obvious that this article cannot discuss any theory in very much detail: the danger is that it could quickly degrade into a completely unreadable mess. Mtroffaes (talk) 13:23, 12 July 2010 (UTC)

Creation of other articles
I also think this article needs an overhaul. However, to not mix up different concepts, we better add some other articles as well; first of all one entitled "imprecise probability theory". Also an "indeterminate probabilities" article could be useful, allowing links between all these articles. Others will present itself once we get up to speed on getting some good pages in wikipedia about this field. Equaeghe (talk) 14:49, 23 September 2009 (UTC)


 * The current section on Walley's interpretation could perhaps serve as a good starting point for an article on lower and upper previsions? (The section feels out of place in this article anyway, at least in its current form.) Mtroffaes (talk) 14:04, 12 July 2010 (UTC)

I think the article is much improved now ... thanks whoever did the most recent efforts... I was also thinking that a pointer to probability theory and philosophy of probability is useful because obviously imprecise probability inherets these problems...  I recently noticed that Walley has an article on the use of imprecise probability in a frequentist setting... but mostly he holds to the use of lower previsions... —Preceding unsigned comment added by 137.194.233.75 (talk) 11:28, 18 August 2010 (UTC)

I made a few edits which I hope are useful... these are somewhat biased by my areas of knowledge which are better for the de Finetti style approach ok for Walley and sketchy for Dempster-Shafer. My overall motivation is to note that there are multiple motivations for the use of both probability and imprecision but to direct the discussion in the de Finetti/Walley direction because I think that is the most inclusive and best developed. I put into some broken/red links to topics that I think Wikipedia should include and maybe I will work on them one day... I also should really add a link to Frank Lad's book because it is really good and really relevant. David Rohde —Preceding unsigned comment added by 137.194.233.75 (talk) 12:07, 25 August 2010 (UTC)


 * Thanks, this is useful! I apologize in case you feel that I seemingly deleted perhaps rather large parts of it, but I did try to integrate as much as possible of it (particularly regarding motivation) into the article. I don't think this is the right place to discuss the interpretation of imprecise probability. What you wrote would probably be more useful in an article addressing lower and upper previsions specificly (unfortunately this still has to be created). There are more interpretations (including frequentist!) of imprecise probability, not just the one that Walley and others have argued for, so it seems rather biased to me to spend a long discussion of Walley's subjective interpretation without mentioning any of the others. Mtroffaes (talk) 17:08, 16 March 2011 (UTC)


 * OK, I take the point, but I think maybe a few worthwhile things got lost along the way.... I think there is value in stating that imprecise probability is a mathematical theory generalizing probability theory but its use in applications requires an intepretation to be placed both on probability and the imprecisison... I am not sure it is possible to avoid discussion of intepretation and in particular the statement: "and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify. Thereby, the theory aims to represent the available knowledge more accurately." seems to be silently adopting an intepretation (as I read it as subjectivist... correct me if I am wrong..).


 * I think the previous opening statment about imprecise probability being a mathematical theory was therefore less biased. I take the point that the de Finetti, Lad, Walley (etc) intepretation might be just one view.  However I wonder if it is possible to talk about applications at all without intepretation, surely this would restrict this article to be about the mathematical topic only?...  BTW Walley also wrote about frequentist imprecise probability (I haven't read this article though).  "Towards a Frequentist Theory of Upper and Lower Probability" Peter Walley and Terrence L. Fine


 * As I understand, from the subjectivist view the main reasons are to model the inarticulateness of an individual or to model the inter-subjective probabilities of a group. I have some reservations about "when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify" as there maybe more sources of inarticulateness than just these.  A very important source of inarticulateness is simply that the partition over which the probability must be specified might be _very_ large, in such a case it would be impractical to elicit subjective probabilities for every possible event.


 * Being slightly pedantic, I also have reservations about "represent available knowledge more accurately". I think a fully specified probability represents an idealization and that imprecision attempts to remove this idealization.  I don't think it is wrong to say this is "more accurately", but I think there may be a clearer word or phrase....


 * With all this said I completely appreciate that you are an expert in this field (and I am not) and in a sense willing to defer to your expertise, but on the other hand I think I make some valid points here....David Rohde —Preceding unsigned comment added by 137.194.233.21 (talk) 14:10, 5 April 2011 (UTC)

Copyright violation
A non-neglectable part of the article, including whole sentences, comes from the article "Imprecise Probability" at the International Encyclopedia of Statistical Science, published by Springer. See http://www.eng.nus.edu.sg/civil/REC2010/documents/imprecise_probabilities.pdf or http://link.springer.com/referencework/10.1007/978-3-642-04898-2/page/13#page-1

On a different note, the article makes subjective unsupported claims like 'the theory has gathered strong momentum', 'concepts ... have become very popular'. In particular, the last claim comes with two references, none of which claims that those concepts have 'become very popular'. In any event, they are from 1967 and 1976, so they could hardly provide any evidence as to the current popularity of those concepts.156.35.192.3 (talk) 16:00, 16 January 2014 (UTC)


 * Thanks for highlighting this. I (Matthias Troffaes) co-wrote parts of the encyclopedia article as second author, and edited this wikipedia article around the same time. The overlap occurred in discussion with my co-authors on the encyclopedia article. I have it on record from my co-authors that they did not think there was a copyright problem in using small bits of the encyclopedia article for the wikipedia article, i.e. they certainly all agreed. However, I'm not a copyright specialist, so perhaps this ought to be reviewed? If there is a violation, I'd be happy to rewrite the relevant sentences to fix the problem.


 * Concerning your other comment, the references do not pertain to popularity at all (as you noted), but merely to the origins of the theory of belief functions. The popularity claim could easily be removed. In fact, perhaps it should be, it is a very subjective notion and does not really add much to the article, certainly not without further objective substantiation. Mtroffaes (talk) 12:04, 24 March 2014 (UTC)


 * I've chased up the copyright form, and unfortunately my co-author who dealt with the copyright form no longer has it on record. Assuming Springer have not substantially changed their copyright form in recent years, their recent copyright forms such as ftp://ftp.springer.de/pub/tex/latex/ccis/copyrightccis_CH.pdf explicitly say that the content can be used for educational purposes, provided that the original source is mentioned. I will add a reference to the original article now. Mtroffaes (talk) 10:07, 15 April 2014 (UTC)


 * Reading up on wikipedia's copyright policy, I'm now somewhat in doubt whether adding a link is enough, because Wikipedia requires the license to be CC-BY-SA-compatible (even though Springer would allow its use for educational purposes). It seems safer simply to rephrase the relevant sentences of the article. Mtroffaes (talk) 10:30, 15 April 2014 (UTC)

Add "Sets of Probability Functions" to Mathematical models
I would suggest adding a sets of probability functions bullet to the mathematical models as it doesn't seem to be subsumed by any of the other suggestions. I thought that was a pretty standard way to talk about imprecise probabilties. E.g. as is presented in http://plato.stanford.edu/archives/win2014/entries/imprecise-probabilities. --Catrincm (talk) 16:21, 6 February 2015 (UTC)
 * It makes a lot of sense indeed. I've added it. Mtroffaes (talk) 10:55, 15 June 2020 (UTC)

Incomplete Probability Preörderings
An editor might want to look at “Formal Qualitative Probability” by Daniel Kian Mc Kiernan, forthcoming in The Review of Symbolic Logic. The prior published work that this article most closely resembles on a formal level is that of Koopman. But it does not impose a logicistic interpretation (nor any other), is not constructed to the purpose of supporting quantification, and is more complete. —184.183.175.18 (talk) 22:20, 20 February 2020 (UTC)

Somewhat Misleading Passage on Keynes
At present, the entry says:
 * In the 1920s, in A Treatise on Probability, Keynes[11] formulated and applied an explicit interval estimate approach to probability.

By his own admission, Keynes's discussion of intervals adds nothing analytically to that by George Udny Yule (“On the Theory of Consistence of Logical Class-Frequencies” in Philosophical Transactions of the Royal Society. Series A, Containing Papers of a Mathematical or Physical Character v 197 [1901] and An Introduction to the Theory of Statistics [1911]). So, at the least, Yule should be mentioned, and prior to Keynes.

(Yule's interpretation is somewhat different. He imagines probabilities as the frequencies with which propositions are true, whereas Keynes imagines them as degrees of plausibility.)

Further, in the case of Boole, of Yule, and of Keynes, all intervals are either the unit interval [0,1] or derived by multiplying the unit interval by a point probability (possibly with some further arithmetic operations). Yet it is clear that Keynes means elsewhere to discuss non-numeric probabilities of a broader class. There is a vociferous crackpot who in vanity journals and in self-published books attempts to argue that Keynes meant everywhere to refer to intervals, but it seems that the first researcher to understand how to treat that broader class also in terms of intervals was Bernard Osgood Koopman (“The Axioms and Algebra of Intuitive Probability” in The Annals of Mathematics series 2 v 41 [1940]). —172.58.22.254 (talk) 00:27, 24 March 2020 (UTC)


 * I'm not personally familiar with the work of Yule, but yes, I agree that if it is relevant, a reference and mentioning of the work should be added. Mtroffaes (talk) 11:00, 15 June 2020 (UTC)

Dead Link / Unmaintained Article
The call-for-papers amongst the external links is quite dead. More generally, this article now seems to be unmaintained. —172.58.19.142 (talk) 01:13, 18 April 2020 (UTC)
 * Thanks for spotting. I've removed the link. Mtroffaes (talk) 10:50, 15 June 2020 (UTC)

Problematic Wording
This passage
 * So, the term imprecise probability—although an unfortunate misnomer as it enables more accurate quantification of uncertainty than precise probability—appears to have been established in the 1990s

has at least three problems. The smallest comes first; it is the “So” ex nihilo. A more important problem is the confusion of the word “precise” with “accurate” or with “exact”. A thing may be infinitely precise and yet grossly inaccurate. And there is the further problem that “imprecise probability” includes conceptions that are not quantifications at all. Better wording would be
 * The term “imprecise probability” is somewhat misleading in that precision is often mistaken for accuracy, whereas a imprecise representation may be more accurate than a spuriously precise representation. In any case, the term appears to have become established in the 1990s

though of course the passage might be revised in other ways to the same end. —172.58.19.153 (talk) 02:43, 29 April 2020 (UTC)
 * These are good suggestions. If there's no objections from others, happy for this to be rephrased in this way. Mtroffaes (talk) 10:49, 15 June 2020 (UTC)


 * How can "conceptions that are not quantifications at all" be more or less precise or more or less accurate? If there are such conceptions, it seems that the text does not yet deal with them. illywhacker&#59; (talk) 14:33, 10 July 2021 (UTC)
 * Given that between two events or propositions one of the relations supraprobability, infraprobability, equiprobability, or non-comparability holds, the imposition of any other of these relations is less accurate. Any point-value quantification imposes one of three relations, supraprobability, infraprobability, or equiprobability.  Thus, all quantifications are less accurate when non-comparability holds, and many quantifications will be less accurate when they impose the wrong ordering on things that are completely ordered.  Accuracy itself is fundamentally an ordering relation, not necessarily quantifiable.
 * The article does barely touch on the aforementioned alternate conceptions. They should instead be front and center.  For a discussion of such conceptions, see the articles by Koopman and by Mc Kiernan to which I refer in my other comments. (Note, there is a transcription error in (A6) in the article by Mc Kiernan. This error is corrected in various places, including an erratum note at ResearchGate.net.)
 * —68.105.244.163 (talk) 22:59, 27 September 2021 (UTC)
 * The reason to put them front-and-center is that they are pure expressions of the most general form of imprecise probability. Intervals can proxy incomplete preorderings, if a mechanism is added to distinguish equiprobability from non-comparability.  But either these intervals are obscuring, or they introduce features that reduce generality. (Again, see Walley's article “Towards a unified theory of imprecise probability”, or just attend carefully to his big book.) —68.105.244.163 (talk) 01:38, 29 September 2021 (UTC)

Sketch of a structure for new version
Hello, I will probably spend a little of my free time updating this page over the next few weeks. I thought I'd share a plan for how I imagine the page looking once it's finished.


 * 1) Introduction
 * 2) History
 * 3) Lower and upper probabilities (basic properties)
 * 4) Connection to other models (credal sets, DS belief functions, etc)
 * 5) Interpretation

I started adding some detail to the history section today, but I hope to do more. Scmbradley (talk) 16:26, 19 October 2020 (UTC)


 * You are proposing to turn the entry into an article about interval-based approaches, and to treat other approaches as something of an afterthought. This would be fundamentally wrong.  In the absence of very strong assumptions that do not represent full generality (Walley, “Towards a unified theory of imprecise probability.”, International Journal of Approximate Reasoning 24.2-3), intervals are just proxies — “a matter of affixing quantitative nails so that an arithmetic hammer may be used” (Mc Kiernan, “Formal Qualitative Probability”, The Review of Symbolic Logic). —72.196.168.48 (talk) 23:59, 20 November 2020 (UTC)

"by George Boole,[3] who aimed to reconcile the theories of logic (which can express complete ignorance)"
This doesn't make sense on its face. Normal logics cannot retain validity in the face of ignorance. If you are talking about a specific logic or group of logics introduced by Boole, you should improve your wording to make this clear. Comiscuous (talk) 07:24, 2 July 2021 (UTC)


 * This is a correct point, and I have edited the article to remove the phrase in parentheses. A revert should add further text explaining how this makes sense, when, on the face of it, logic deals with certain propositions. illywhacker&#59; (talk) 14:31, 10 July 2021 (UTC)

Poposed Change of Title and Rewrite
While references (Walley, Koopman, Mc Kiernan) have been provided on this talk page, it seems clear that no one is both willing and able to write a proper article about generalized imprecise probability. Instead, this article remains conceptualized in terms of intervals. Perhaps someday someone will be willing to construct a more general article, but for now the article misleads its readers.

My suggestion, then, is that the article be retitled “Interval-valued probability”, and rewritten to be a presentation of just that, with passing mention of incomplete preorderings as the most general notion of imprecise probability. —68.105.244.163 (talk) 01:37, 21 January 2023 (UTC)