Who thinks probability is just a number? A plea.

Many people think – perhaps they were taught it – that it is meaningful to talk about the unconditional probability of ‘Heads’ (I.e. P(Heads)) for a real coin, and even that there are logical or mathematical arguments to this effect. I have been collecting and commenting on works which have been – too widely – interpreted in this way, and quoting their authors in contradiction. De Finetti seemed to be the only example of a respected person who seemed to think that he had provided such an argument. But a friendly economist has just forwarded a link to a recent work that debunks this notion, based on wider  reading of his work.

So, am I done? Does anyone have any seeming mathematical sources for the view that ‘probability is just a number’ for me to consider?

I have already covered:

There are some more modern authors who make strong claims about probability, but – unless you know different – they rely on the above, and hence do not need to be addressed separately. I do also opine on a few less well known sources: you can search my blog to check.

Dave Marsay

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About Dave Marsay
Mathematician with an interest in 'good' reasoning.

15 Responses to Who thinks probability is just a number? A plea.

  1. Blue Aurora says:

    What about George Boole?

  2. One more thing comes to my mind which could be relevant:

    Physicist Scott Aaronson argues at http://www.scottaaronson.com/democritus/lec9.html that the core of Quantum Mechanics is just a generalization of probability theory.

    He further argues: “if you want a universe with certain very generic properties, you seem forced to one of three choices: (1) determinism, (2) classical probabilities, or (3) quantum mechanics”

    So one might argue:

    1.) Probability Theory is a generalization of logic (as argued by Jaynes, see also http://bayes.wustl.edu/ )
    2.) Quantum Mechanics is a generalization of probability theory

    • Dave Marsay says:

      Thanks. Scott sheds some light on the mystery that is QM. I get that the classical universe (that we see) is just a projection of some more complicated universe, much in the same way as classical probability might be a projection of some more complex uncertainty. But it seems to me that most QM (as in the Copenhagen interpretation) applies classical probability and tries to gloss over any problems. David Deutsch seems to me to be an exception. One problem is that the Bayesian updating rule is non-relativistic, so one has a clash between relativity and QM. DD writes around this, and it seems to me that some of his work can be read as supporting the view that weight of evidence is a more fundamental (and reliable) concept than classical probability / information (which only make sense within epochs), and so QM seems to me to demand a reformed concept of probability. But, as far as I know, physicists still seem to calculate classically and then have various rules of thumb to work around any problems that this might cause, so they are pushing for any reforms. Of course, their advantage is that they can do experiments, and so do not need to rely on this aspect of mathematics is not. But I do nudge any that I meet, and enquire about any change.

  3. Bob Nease says:

    I trained at Stanford under Ron Howard. He was very, very careful to note that there is no such thing as an unconditional probability; his preferred syntax for the probabilit of x was {x | E}, in which E represented all of the experience of the decision maker (or the decision maker’s agent) at the time of the probability assessment. This was back when formulas were written on paper with mechanical pencils, and the act of conditioning all of the assessments created muscle memory of the importance of this concept.

  4. Mr.Marsay I have never read the classics outside of having Jayne’s on my bookshelf is there any book you would recommend on probability before all else? It’s going to take me a long time to work through all that I have. What reading order would you recommend?

    • Dave Marsay says:

      SE, I had a quick look at your reading list and saw little that I had read myself. You seemed to have summary works rather than original work. In case your reading seems to give any authorities for the view that probability is just a number could I ask you to consult my blog for any contrary quotes, and let me know if you come across an argument that I haven’t covered?

      As for suggesting a book, it seems very much to depend on ‘where you are coming from’. If you found something you liked I may be able to suggest something that you might also like to read. But there is no general book.

      • “SE, I had a quick look at your reading list and saw little that I had read myself. You seemed to have summary works rather than original work. In case your reading seems to give any authorities for the view that probability is just a number could I ask you to consult my blog for any contrary quotes, and let me know if you come across an argument that I haven’t covered?”

        Yeah sure will do.

  5. Ian Chafe says:

    Have any of you ever heard the names Andrey Kolmogorov or Norbert Wiener? They are the founders of modern mathematical probability theory. Kolmogorov in particular laid the mathematical foundations of pure probability with something called measure theory; if you are a mathematician then you will have necessarily studied it. Wiener in his rigorous work on Brownian motion as a measure in function space (developed further by Paul Levy, PhD advisor of Benoit Mandelbrot) along with Kolmogorov also laid the mathematical foundations for stochastic process theory. You should be addressing these people if you are serious about probability theory.

    • Dave Marsay says:

      Thanks for your comment. If you search my blog for Kolmogorov you will see my comments on his ‘foundations’. Basically, l get that by analogy we should expect upper and lower measures, but I have not found an argument by Kolmogorov that these measures must always be the same. Similarly, although I have read some of Weiner, Levy and Mandelbrot I have not found any argument by them that I must be a stochastic process. Please could you let me have any specific references? There is a strong probability that I may reply by blog 😉

      Thanks again.

  6. David Harold Chester says:

    Probability and macroeconomics of our social system don’t mix! Many people claim that they do and the result is confusion because there is a lot of complex behavior that should be sorted out first, and quite often it isn’t.

    I don’t see how one can seriously study macroeconomics until this complexity problem is solved first, but after the solution (which I have shown in SSRN2865571 “Einstein’s Criterion Applied to Macroeconomics Modeling”) has been determined, its very clear that the aggregates are all 100% probable and no alternatives exist (or they have zero probability)! Those economists who insist on probability must surely mixing up micro- and macro- , which a part of the confusion.

    The following poem should prove helpful; so as to better appreciate what is involved. The author was Kenneth Boulding, (1910 – 1993).

    A system is a big black box
    Of which we can’t unlock the locks
    And all we can find out about
    Is what goes in and what comes out.

    Perceiving input-output pairs,
    Related by parameters,
    Permits us, sometimes, to relate
    An input, output, and a state.

    If this relation’s good and stable
    Then to predict we may be able,
    But if this fails us heaven forbid!
    We’ll be compelled to force the lid!

    Logic followed by lid-opening “persuasion” are the answers!

    • Dave Marsay says:

      I have had a small peek at what’s under the lid of the UK system, and there are many people who act as if they think that the system is stable, while others work hard to keep it so. They can succeed in what they think are the important aspects quite well, only to find some new factor emerging. Sometimes they can catch these too, but not always.

      So far I have only read your abstract. Some such modelling work seems essential. Possibly in practice we might judge that it would be in everyone’s interest to sustain such a system, so ‘pragmatically’ it might be stable. But if we did ever have such a system I think someone ought to be looking out for exploitable weaknesses and groups with potential motivations to ‘shake the tree’. But maybe I should read the paper!

  7. davetaylor1 says:

    Great “Boulding” comment from David Chester! The thread on Bayes seems to scupper my belief that Bayes’ theorem is (in Knightian terms) assessing uncertainty given risk (or vice versa), but in fact the meaning of the theorem does not arise in your quotations, while known options are acknowledged by Bayes in his explicitly leaving them out.

    Unconditional probability as a number is an interesting question. The concept of a number I take to be a set of distinct units, which could be instanced by two coins but not by two sides of one coin. (If when tossed that landed on soft ground it could end up stuck on edge: neither one side nor the other). The probability of picking up a heads coin rather than a tails seems to involve not facts but possibilities, i.e. not deductive but modal logic. It seems to me the facts which turn up in coin-tossing or whatever can only be mapped to probabilities given rules defining “proper” which ensure the independence of events.

    • Dave Marsay says:

      Whatever the details, a key point in thinking about the probability of ‘Heads’ for a real coin is that you either end up with a circular argument or some belief that you just accept on faith or which you think – at least as a matter of principle – requires further justification. Hence irreducible uncertainty. But we still need people like Chester to reduce the uncertainty as far as we can at the time given … .

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