Probability:-

An overview of the key works on probability (let me know if you disagree). Dale A History of Inverse Probability: from Thomas Bayes to Karl Pearson provides an alternative overview with selected quotes. But it seems to me that he understates the authors’ reservations about the generality of their theories.

Classics

Bayes and Laplace are often regarded as the founders of probability theory.

Bayes develops a numeric theory of probability, based on utility, including Bayes’ rule, and shows how to infer a probability interval. But he also cautions about extrapolating too far.

Laplace developed a version of Bayes’ rule based on utility, but recognizing that utility is typically ‘sub-linear’ or convex and that one should take account of any specific evidence that may conflict with, for example, Bayes’ rule. Also, Laplace recognizes that short-term regularities may not always be continued indefinitely.

Ramsey is often regarded as providing the mathematical foundations of subjective probability, based on ‘mathematical expectation’, but notes the limitations.

Jeffreys developed an ‘objective’ approach, based on selecting the ‘simplest’ of possible explanations.

Savage developed an alternative approach, and was almost evangelical in promoting the likelihood principle and ‘Bayesian’ approach for scientific experiments. (But see Binmore‘s comments.)

Fisher notes some severe restrictions on the justifications for probability theory. (These are perhaps not always appreciated as they might be.)

Other works

Jaynes develops a probability theory based on entropy maximization. He notes that this implicitly assumes ‘an infinitely educated brain’.

Lindley is a well respected advocate of ‘Bayesianism’, but emphasises ‘the need to think’.

Keynes developed general, non-numeric, theories of probability, which he applied as the basis of his economic theories. Binmore has developed a theory based on the notion of ‘muddled’ sequences, as a generalization of conventional random sequences. Neither are well known.

Russell reviewed the then(1940s) mainstream theories, adapting Keynes’. He showed that conventional (numeric) probability was justified when normal science was, and discussed what these conditions might be. Perhaps confusingly, Russell uses the term ‘mathematical probability’ for the run-off-the-mill theory, only appropriate to run-of-the-mill circumstances. But it seems to me that the different theories espoused by Keynes in his Treatise for his mathematics fellwoship are also ‘mathematical’, albeit in need of a little development.

See Also

  • Related material is under Economics, e.g. where Bernoulli , de Finetti and von Neuman & Morgenstern argue from economics to probability, or to the limits of probability theory.
  • Many works under Rationality and Uncertainty argue for non-measureable uncertainty.
  • Some works on ‘how people actually reason’, such as Ellsberg‘s, are also of interest.
  • My blog has a browsable index, using mouseover. Pages on probability can be found under ‘bibliography/rationality and uncertainty/probability’. Sadly, it doesn’t work on most smartphones (2012).

Dave Marsay

7 Responses to Probability:-

  1. Blue Aurora says:

    Out of curiosity, Dr. Marsay…have you ever read anything by the late American mathematician Morris Kline? If you have not heard of him or read anything by him, I suggest picking up a copy of this book by him.

    http://www.amazon.com/Mathematics-Loss-Certainty-Galaxy-Books/dp/0195030850/

    If you want a lengthier (but still scholarly) treatment of the history of mathematics by the same author, you may wish to read Mathematical Thought from Ancient to Modern Times, which can be bought as a three volume work.

    • Dave Marsay says:

      I’ve not seen this book. I shall enquire of my mathematics teacher friends. From the bibliography and index it seems silent on probability, which is a shame. From the introduction its main point seems to be that mathematics is not what it was thought to be prior to the 20th century. Agreed. Conceptually, mathematics as it is now understood hardly existed 100 years ago. But I didn’t find an account of what mathematics actually is, which seems to me much more important than the history.

      (For example, Hilbert’s Geometry is ‘good mathematics’ irrespective of the what delusions people once had about it.)

      More thoughts along this line can be found at https://djmarsay.wordpress.com/bibliography/rationality-and-uncertainty/mathematics/blackwell-philosophy-of-mathematics/ .

      • Blue Aurora says:

        I see. However, I don’t think you would deny that a knowledge of history is an important part to helping one understand the purpose of a subject. Then again, I understand that your comment was based off a cursory glance. Had you heard of Morris Kline before I mentioned him in my comment? I have yet to read anything by Morris Kline for myself, but I mentioned him because I thought his work might be interesting to you.

        But speaking of probability…have you heard of a mathematician named James Franklin? I believe he is a tenured professor at an Australian university. He is the author of The Science of Conjecture: Evidence and Probability before Pascal, which was published by Johns Hopkins University Press in 2001. You may also find James Franklin’s book pertinent to your interests.

        http://www.amazon.com/The-Science-Conjecture-Evidence-Probability/dp/0801865697/

      • Dave Marsay says:

        One certainly needs to understand some history to understand economics, and similarly for ‘mathematics as a tool’. But I have yet to find a good book on mathematics as critical reasoning. My knowledge of mathematicians is largely confined to Brits. Understanding them is hard enough!

      • Blue Aurora says:

        I apologise for my belated response, as real life considerations had to take place first. I’m glad to hear that both of us agree on the importance of historical backgrounds. As for finding a good book about mathematics as critical reasoning…well, perhaps Morris Kline’s book might be the book for you. Perhaps it isn’t. But either way, I think you ought to read it. As for James Franklin (who is native to Australia) and non-British mathematicians…I see. Of course, a scholar’s nationality (ideally) shouldn’t get in the way of their arguments.

  2. Probably Polya for critical reasoning Mr.Marsay

    • Dave Marsay says:

      Polya used to be recommending reasoning at school. Some people seem to suppose that reasoning is formulaic, such that ‘logic’ resembles computer logic. Boole that effective reasoning has two modes. One is the heuristics that he discusses at some length. They give ‘candidate solutions’. The other mode is to check and refine candidates into proven solutions. Typically one has to iterate between these modes.

      He also says something about uncertainty which I should re-read. Thanks.

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