The Quote of the Week: Lyons on Single Event Probabilities
    By Tommaso Dorigo | January 10th 2013 09:57 AM | 7 comments | Print | E-mail | Track Comments
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    I am an experimental particle physicist working with the CMS experiment at CERN. In my spare time I play chess, abuse the piano, and aim my dobson...

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    "Given that a repeated series of trials is required, frequentists are unable to assign probabilities to single events. Thus, with regard to whether it was raining in Manchester yesterday, there is no way of creating a large number of `yesterdays' in order to determine the probability. Frequentists would say that, even though they might not know, in actual fact it either was raining or it wasn't, and so this is not a matter for assigning a probability. And the same remains true even if we replace `Manchester' by `the Sahara Desert'.

    Another example would be the unwillingness of a frequentist to assign a probability to the statement that `the first astronaut to set foot on Mars will return to Earth alive.' This does not mean it is an uninteresting question, especially if you have been chosen to be on the first manned-mission to Mars, but then, don't ask a frequentist to assess the probability."

    Louis Lyons, "Bayes and Frequentism: a Particle Physicist's Perspective"


    As someone who is a bit acquainted with rocket technology via a parent, I don't think a frequentist would have any trouble assigning a probability of various forms of mission failure on a mission to Mars. The more detailed the engineering specs, the more detailed the failure probability and better yet, conditional failure probabilities can be described. It all boils down to being able to model the mission, and Monte Carlo simulate it.

    You can assign probabilities to known or potential failures. However, as anyone who has done something new, it is the unknown or not thought of potential failures that can't be assigned probabilities and therefore make such a one time, first event unable to have probabilities assigned. A lot of times these unknowns do crop up.

    Nobody, not frequentists, not Bayesians, can assign probabilities accurately to unknowns. In Monte Carlo simulations, all you can do is use probability distributions with long tails to failures of components or to forcing functions, so that the failure modes induced by unforeseen events are "taken care" of. E.g., if the aerodynamic forces acting on the spacecraft descending and ascending from Mars have some unknowns relative to aerodynamics in Earth's atmosphere, include such by adding some probability for unusual aerodynamic forces in the simulations.

    Tommaso says that frequentists can't even philosophically grasp this situation, and that is not true.

    > I don't think a frequentist would have any trouble assigning a probability of various forms of mission failure ... the more detailed the engineering specs, the more detailed the failure probability

    Clearly, someone is sitting in the wrong church here.

    Additionally, you don't get "failure probabilities" from engineering specs.

    Talking of which, during the Challenger Space Shuttle Launch Decision: the subjective estimate of catastrophic failure of the booster was totally off. Not because examinations revealed that the booster was always a hair's breadth away from behaving badly (as they did) but because its repeated failures to behave catastrophically induced the idea that things would work out next time too (the error seems to be that if you get N times a six when throwing a die, you conclude that the next time will be a six too; instead of concluding that you have been very lucky; is this "frequentism" or just "having a bad model"? The human tendency to fit facts to convenience - in this case, a booster redesign would have meant politically unacceptable program interruptions - will worsen things. Exactly the same happened for Columbia btw.

    >> assign a probability to the statement that `the first astronaut to set foot on Mars will return to Earth alive.'

    I just assigned the probability of 12.345% to this unique event.
    I hope this helps ...

    Dear Tommaso,

    I´m surprised to see that the article do not mention Cox´s theorem, which to me is one of the most important result on the subject of the last century.
    Very interesting, and worth mentioning also are the works of Jaynes.
    It is important to say that Bayesian, or subjective, does not mean arbitrary: people must be coherent and reasonable!

    What do you think about it?


    So ... Cox's 1946 theorem is an axiomatization of the calculus of probability.

    But it's not the first!

    From a href="">The origins and legacy of Kolmogorov's 'Grundbegriffe':

    In an article published in Russian in 1917 and listed by Kolmogorov in the Grundbegriffe’s bibliography, Sergei Bernstein showed that probability theory can be founded on qualitative axioms for numerical coefficients that measure the probabilities of propositions.

    Bernstein’s two most important axioms correspond to the classical axioms of total and compound probability:

    • If A and A1 are equally likely, B and B1 are equally likely, A and B are incompatible, and A1 and B1 are incompatible, then (A or B) and (A1 or B1) are equally likely.

    • If A occurs, the new probability of a particular occurrence α of A is function of the initial probabilities of α and A.

    Using the first axiom, Bernstein concluded that if A is the disjunction of m out of n equally likely and incompatible propositions, and B is as well, then A and B must be equally likely. It follows that the numerical probability of A and B is some function of the ratio m/n, and we may as well take that function to be the identity. Using the second axiom, Bernstein then finds that the new probability of α when A occurs is the ratio of the initial probability of α to that of A. Bernstein also axiomatized the field of propositions and extended his theory to the case where this field is infinite. He exposited his qualitative axioms again in a probability textbook that he published in 1927, but neither the article nor the book were ever translated out of Russian into other languages. John Maynard Keynes included Bernstein’s article in the bibliography of the 1921 book where he developed his own system of qualitative probability. Subsequent writers on qualitative probability, most prominently Bernard O. Koopman in 1940 and Richard T. Cox in 1946, acknowledged a debt to Keynes but not to Bernstein. The first summary of Bernstein’s ideas in English appeared only in 1974, when Samuel Kotz published an English translation of Leonid E. Maistrov’s "history of probability".

    Unlike von Mises and Kolmogorov, Bernstein was not a frequentist. He had earned his doctorate in Paris, and the philosophical views he expresses at the end of his 1917 article are in line with those of Borel and Levy: probability is essentially subjective and becomes objective only when there is sufficient consensus or adequate confirmation of Cournot’s principle.

    Btw. how do I suggest the site manager (Hank?) do the following:

    Make the commenting boxes wider. 60 characters width is pre-VT220
    Give the text boxe for captcha copypasta a specific background color. On my LCD screen I have to look at it from a large angle to even see it. Yes, I threw out my gigantic glasstube.