Re: More Hard Problems Using Bayes' Theorem, Please

From: Eliezer S. Yudkowsky (sentience@pobox.com)
Date: Wed Jul 09 2003 - 01:09:45 MDT

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    Lee Corbin wrote:

    >>> The probability that a newborn will have deformities
    >>> traceable to a sickness of its mother during pregnancy is 1%.
    >>>
    >>> If a child is born healthy and normal, the probability
    >>> that the mother had rubella during her pregnancy is 10%.
    >>>
    >>> If a child is born with deformities and it can be traced
    >>> to some sickness of the mother, the probability that the
    >>> mother had rubella during her pregnancy is 50%.
    >>>
    >>> What is the probability that a child will be born with
    >>> deformities if its mother had rubella during her pregnancy?

    The last question, if read literally, allows for the possibility of
    deformities not caused by rubella. Similarly the second statement would
    have been better written "If a child is born without deformities traceable
    to a sickness of its mother". Personally I found it quite evident what
    was meant; but yes, it is not a perfect phrasing.

    > Oh, all right. It just looks as though they are saying p(HD) != P(H)*p(D|H):
    >
    > The most frequent non-Bayesian algorithms they identified
    > include computing p (H&D) by multiplying p (H) and p (D | H);
    >
    > because they are giving points to the test-takers only for the
    > use of Bayes formula---and, apparently, only to using the
    > posterior/prior odds formulation of it no less---(and in a
    > faulty problem to boot).

    I do not think the authors of this paper are investigating what you think
    they are investigating. They are investigating failures of rationality
    that everyone, Bayesians and frequentists alike, agree upon. They simply
    call them "non-Bayesian" which is a standard term for "wrong answer" *in
    the academic subfield of heuristics and biases*, regardless of its meaning
    in the academic subfield of statistics.

    Offhand I cannot recall a case of Gigerenzer getting mixed up in the
    dispute over Bayesian versus frequentist methods. (Note use of
    availability heuristic...) He works in the Tversky-and-Kahneman,
    heuristics-and-biases field, where it is uncontroversial to equate
    Bayesian reasoning with normative rationality and "the Bayesian answer" is
    simply a more technical way of saying "the correct answer" or "the
    normative answer".

    They are giving points to the subjects only for the correct answer,
    however arrived at - they do not check the work done to see if Bayes'
    Theorem was explicitly used.

    > Yes, I've examined his book on-line, but then they took it offline
    > while the book was being printed. Last time I checked, it was back.
    > But the book was supposed to be published this year (all 2000 pages
    > or whatever of it), and that's what I want to get.
    >
    > I do need better examples of where the Bayesians and non-Bayesians
    > disagree. (I have wanted to be in on that controversy since 1972,
    > and always felt side-lined.) I hope that not all Jaynes' examples
    > have to do with prior distributions, unbiased estimators,
    > transformations of likelihood, and so on.

    Not the book, "Probability Theory, the Logic of Science". The lectures,
    "Probability Theory with Applications in Science and Engineering", which
    are online, and also more accessible.

    http://bayes.wustl.edu/etj/science.pdf.html

    -- 
    Eliezer S. Yudkowsky                          http://singinst.org/
    Research Fellow, Singularity Institute for Artificial Intelligence
    


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