Brady+ on Keynes’ Interval Probability

Rogerio Arthmar & Michael Emmett Brady Keynes’ lower-upper bound interval approach to probability.

(To appear in the History of Economic Ideas as “Boole, Keynes, and the interval approach to probability”.)

… Keynes provided a complete mathematical structure for his system of probability … he provided a solid mathematical structure for his … approach. We conclude that Keynes should be recognized as the founder of the modern non additive approach to probability. 

This is ostensibly an historical paper, correcting the misconceptions of some long-dead mathematicians and economists, and of Skidelsky. Its importance is that, as in Keynes’ day, many ‘practical men’ are still bamboozled by such misconceptions, as evidenced by the financial problems of 2007/8 and the lack of progress since.

I read the main text as a mathematician with a different background to what I assume is the intended readership. But some key points jump out. While I agree that Keynes’ work has not been given full credit, this paper – it seems to me – continues to understate the position. As a mathematician, I see Keynes in the context of his tutor, Whitehead, his fellow student Russell and those who followed, including Turing and (Jack) Good. Thus while Keynes’ work predates the proper foundation that Russell and Whitehead, and Turing, helped to develop, and so is not fully developed and even formally deficient, it is counted by both Whitehead and Russell as contributing to their ideas, and hence – I would argue – should be considered ‘properly mathematical’ in conception, if flawed in presentation. Thus it seems to me that the paper is misguided in trying to use  inadequately mathematical works as some sort of standard to compare Keynes’ work with. I would rather use Keynes as the standard for them.

I would rather say that Keynes’ Treatise, in conjunction with the applications of Keynes, Whitehead, Russell, Turing and (Jack) Good,still provides the  best  source of understanding of uncertainty. Anticipating Turing’s work, it does not attempt to provide a complete formula or method for every occassion, nor to reduce uncertainty to ‘obvious’ axioms, like those of Geometry. Instead it provides a rich pallete of possibilities from which can be constructed models that can be tested (scientifically) in particular cases. This includes conventional (Bayesian) probability, comparative probability, interval probability, conditional probability and Boole’s probability. Keynes showed that conventional probability was appropriate for what was then conventional scientific practice, based on certain assumptions. The other (non-additive) forms are suitable more generally, but Keynes does not justify their use in particular practical cases.

Thus I would wish to make a considerably larger point than the paper. But the paper does raise some important issues.

Much mathematics is relatively mature, and concerns ‘accepted axioms’ and ‘approved methods’ whose applications can be justified by ‘arguments to authority’. The text seems to assume – as many do – that probability theory is one such area. The main point to arise from even a casual reading of Keynes is that he did not regard it as mature, except in certain areas. Of course, non-mathematicians  tend to only ‘see’  mature applied mathematics, and organisations tend to only want to use ‘established methods’. Hence there is pressure on mathematicians (like other professionals) to present their work as if it is mature when actually it isn’t. But it would be more faithful to Keynes to be more honest. The problem of mathematical finance in the bubble prior to the 2008 crash was – in my view – not so much that the methods were poor but that the significance of the results was misunderstood. Even a superficial reading of Keynes would have helped. I am not sure that the adoption of interval probabilities would have helped, or would help in other areas.

I also disagree with the above abstract on these points, explicit or implicit:

  • Keynes work isn’t really complete, and – particularly at the end – shows clear (but excusable) signs that his ideas, let alone the text, had not fully matured.
  • It would probably be fairer to give some credit to his colleagues, such as Whitehead and Russell. Keynes wrote the ideas up, but I doubt that they were his alone. (Indeed, Keynes cites Whitehead.)
  • Keynes has not provided an adequate text for any ‘school of Keynesian probability’.
  • Keynes would surely wish to be remembered for his logical approach to probability, of which non-additivity is a consequence.

I further note that the paper seems to regard probability as Platonic: as somehow existing ‘out their’, independently of our conception of it. But Keynes seems to argue quite effectively that any particular concept or model that we have of uncertainty will be incomplete, and that we need to invent models to meet particular needs and then to test our model, e.g. scientifically. In my experience this can best be done by working from an understanding of the problem domain. For examples, different notions are appropriate to games and gambles.

See Also

My notes on probability (e.g. Ramsey) and mathematics.

A previous, similar, paper by M.E. Brady.

Dave Marsay

3 Responses to Brady+ on Keynes’ Interval Probability

  1. Blue Aurora says:

    This is an excellent review of Dr. Michael Emmett Brady’s paper. I recommend e-mailing him your comments and criticisms. (Dr. Brady told me that there were going to be four commentators in the issue for which the Boole/Keynes paper is going to appear.)

    But you haven’t seem to have read Theodore Hailperin’s works, Dr. Marsay. I have promised to you before to send you a copy of an article by Theodore Hailperin, but I have unfortunately failed on that. When I go back to university, I shall remedy that by giving you the article in question. However, here are two books by Theodore Hailperin that I believe you ought to read.



    Regarding the first book, there’s a newer printing over here.

    Finally, have you ever heard of David W. Miller? He’s a statistician who passed away two years ago, but he wrote a book on George Boole’s Last Challenge Problem. You can buy and download the book over here.

    Be warned – don’t forget the password for accessing the book, as it is copy-righted.

    • Dave Marsay says:

      Thanks. I’ve made some notes on Hailperin (seach on blog). I agree that it is highly relevant, but what I’ve seen didn’t seem to add much to Jack Good et al. (Unless you could recommend a specific paper?)

      Miller is much more interesting once you get seriously in to challenging areas, but it seems to me that the priority is to motivate an appreciation of imprecise probability in the first place. Again, any suggestions?

      I have LinkedIn to MEB, but we have yet to engage.

      • Blue Aurora says:

        The paper by Theodore Hailperin that I recommend and that Dr. Michael Emmett Brady cites would be this one:

        Hailperin, T. 1965. “Best possible inequalities for the probability of a logical function of events”. American Mathematical Monthly, 72, 343-59.

        Theodore Hailperin’s 1996 book, Sentential Probability Logic, is apparently very good. I’d get it from your library, and from there, devise a way to establish applications to decision theory and the like.

        As for David W. Miller, he is good for challenging areas, but yes, I think the goal is to get an appreciation for imprecise probabilities/interval-valued probabilities. However, I don’t know any academic papers of his that deal with George Boole beyond that book.

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