Why call it Bayes' Theorem?

From: Lee Corbin (lcorbin@tsoft.com)
Date: Wed Jul 09 2003 - 17:28:11 MDT

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    When you are 19, your intuition is clear as a bell, and you
    have utter and supreme confidence in it. You have not seen,
    normally, baffling counter-examples to "obvious" theorems,
    or the two-envelopes problem, Monte-Hall, Simpson's Paradox,
    Unexpected Hangings, etc.

    That's how it was with me in 1967 when I was 19 and first
    encountered Bayes' Theorem in Gnedenko's Theory of Probability
    textbook. I was appalled that something so simple as the Bayes'
    Formula should have a *name*. To me, it was as if someone's
    name were attached to the general solution of N equations in N
    unknowns.

    (This is a phenomenon among the quite young; a few years ago
    a brilliant mathematician I knew---still a teenager---had run
    into the Borel-Cantelli lemma in probability theory, and was
    in turn appalled that something so "obvious" had a name!)

    Only ten or fifteen years ago did I discover the *true*
    problem that the Reverend Bayes worked on, and why indeed
    his solution/theorem warrants the appellation, and why the
    elementary formula also should therefore be named in his honor.

    Here is the sort of problem he had obtained an answer to:

      In the next room you hear shouts of "I win!" alternately
      with "Damn!" from one particular player, and you realize
      that he is playing some sort of game with perhaps cards
      or dice.

      You hear "I win!" three times, and "Damn!" twice. What
      is the probability that he will win the next round?
      (Or, for the non-Bayesians, what do you estimate to be
      the probability or likelihood he will win the next round?)

    I *believe* that the non-Bayesians will calculate a likelihood
    curve, but the Bayesians will go right ahead and announce a
    probability. (With a prior uniform distribution, I, and the
    Reverend Bayes, and the Bayesians---I think---use a little
    calculus to get an answer. I will go dig up my answer and
    post it.)

    Here is Bayes' original problem, in his words, in the 1750s or so:

       *Given* that the number of times in which an unknown event has
       happened and failed: *Required* the chance that the probability
       of its happening in a single trial lies somewhere between any
       two degrees of probability that can be named.

    Already he wanted the probability density (distribution function).
    And in his paper published later by the Royal Society, he got it.

    Lee

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