Saturday, September 17, 2011

More about our friend Bayes

When the facts change, I change my opinion.
What do you do, sir?
John Maynard Keynes

In a review of a new book "The Theory That Would Not Die" (S. McGrayne, May,2011) mathematician John Allen Paulos says this:

"At its core, Bayes’s theorem depends upon an ingenious turnabout: If you want to assess the strength of your hypothesis given the evidence, you must also assess the strength of the evidence given your hypothesis.

In the face of uncertainty, a Bayesian asks three questions:
  1. How confident am I in the truth of my initial belief?
  2. On the assumption that my original belief is true, how confident am I that the new evidence is accurate?
  3. And whether or not my original belief is true, how confident am I that the new evidence is accurate?"

In other words, the Bayesian idea is this:
  1. Form a hypothesis; assess the probability of its being true
  2. Seek evidence that the hypothesis is true
  3. Recalculate the probability of the hypothesis, given the evidence (assuming the hypothesis is TRUE) and the probability of finding the evidence
There are some obvious "calibration" problems, the same sort that show up in the risk matrix, and these calibration issues have challenged the efficacy of Bayesian reasoning in the same way:
  • It's difficult to get a calibrated probability for the initial hypothesis--what's the benchmark here, and can you be sure that biases are removed?
  • It's difficult to find the supporting evidence with clarity of cause and effect
  • It's difficult to assess the probability of finding the evidence--again, there may be no historical basis for such a probability.

Nevertheless, according to McCrayne's history, Bayes has these credits:

"It was used to search for nuclear weapons, devise actuarial tables, demonstrate that a document seemingly incriminating Colonel Dreyfus was most likely a forgery, improve low-resolution computer images, judge the authorship of the disputed Federalist papers and determine the false positive rate of mammograms. She also tells the story of Alan Turing and others whose pivotal crypto-analytic work unscrambling German codes may have helped shorten World War II."

For more, see my prior posts.

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