RE: More Hard Problems Using Bayes' Theorem, Please

From: Lee Corbin (lcorbin@tsoft.com)
Date: Tue Jul 08 2003 - 19:35:16 MDT

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    Dan Fabulich wrote yesterday at 7pm or so

    > But, let's get back to your [Mitchell's] question. There is no hard-and-fast
    > definition of "frequentist" in this debate, unfortunately.

    The likelihood of that sounds high to me. :)

    > > 3. Using orthodox methods and an expensive statistics program,
    > > calculate the probability that, if the hypothesis "I am looking at the
    > > red barrel is correct", you will see 7 or more red tokens in ten
    > > samples. Explain why this number is actively harmful to genuine
    > > statistical understanding in twenty-five words or less.

    > IMO, the relevant "frequentist" position at which Eliezer was poking fun
    > here was the position that there are some common-sense events that have no
    > probability whatsoever; these "singular" events don't have a probability
    > of 0 or 1, they just don't have "probabilities" at all... according to
    > this type of frequentist, the whole notion is simply not applicable
    > sometimes.

    I don't know what his point was, but I don't that that it
    is what you have written. Yes, the only traditional reasons
    separating the Bayesians and the non-Bayesians seems to be,
    as I put it decades ago, "the non-Bayesians are too chicken
    to say what the probability is in some cases".

    > A good example might be to ask what the odds might be that a certain
    > couple will get married. In this case, the frequentist feels obligated to
    > respond that "probability" is a misnomer here: probability only applies
    > when there's a large set of things that can be literally counted. If
    > there were a million pairs of nearly identical couples, THEN the
    > frequentist would let it count as a "probability", but not otherwise.

    Sounds good.

    > Hence, some kind of frequentist might say of Eliezer's question 1, as
    > posed:

    > > 1. After repeatedly revising your probability to take each of these
    > > observations into account, what is your estimated chance that the barrel
    > > is the one containing mostly blue tokens?

    > that the question, as stated, doesn't make any sense: there is no such
    > thing as an estimated probability or prior chance in this case.

    I can't imagine *anyone*---"frequentist" or not---who would fail to
    announce a probability for the barrel in question to be the one
    mostly containing blue tokens. The a priori of 1/2 was clearly
    stated, whereas in your marriage example, of course, it's really
    not so easy to pin anything down, as you explained.

    > But that frequentist WOULD be comfortable answering question 3.

    Namely, I think you are saying, that a frequentist would be willing
    to announce a probability that you will see seven or more tokens
    *given* that you are drawing from the barrel containing mostly red
    tokens.

    > Furthermore, if this frequentist were really hard-core, she might reject
    > Bayes' Theorem altogether and try to find the frequency by actually
    > repeating the experiment a bunch of times. That, the frequentist might
    > insist, would be the only way to guarantee that a meaningful result was
    > being calculated.

    I cannot believe---until someone gives me some evidence---that there
    is anyone who doesn't make free use of Bayes' formula in a case like
    this. They know damn well what frequency they'll get, because logic
    grabs you by the throat and FORCES you to calculate it. ;-)

    > Looking back at the problem, I have no idea which barrel is red, (since
    > neither barrel was painted,) but, on the presumption that the barrel with
    > mostly blue tokens was painted blue, and the other barrel (which you
    > failed to notice had an EQUAL number of blue and red tokens) must have
    > been painted red.

    Yes, Eliezer did not state the problem using absolutely correct
    terminology, but his meaning was clear enough. Why are you picking
    on the possibility of a difference between "the red barrel" and "the
    barrel containing mostly red tokens"? It's clear that this is what
    he meant, right?

    > Neither do I find myself able to say much about how the
    > frequentist, even my hard-core frequentist, has actively harmed her
    > statistical understanding... even frequentists can plug numbers into
    > Bayes as well as anybody else, but sometimes they choose not to on
    > principle.

    Exactly.

    > But if I had to give an answer, it would be this: "orthodox statisticians
    > cannot explain WHY the given answer is correct; they couldn't have
    > predicted the answer, and couldn't therefore have been "surprised" by the
    > results." [Hence, they have left the path of science.]

    Now it seems to me that in simple problems like these that do not
    involve distributions, orthodox statisticians can explain just as
    well as Bayesians why an answer it *correct*. They merely, (so far
    as I know, and as you pointed out) simply refuse to call certain
    estimates "probability", and don't appear to have much use for those
    calculations.

    I'm sure that there is more to it than that, but that's how I see it
    right now.

    Lee



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