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The Mathematics of Changing Your Mind

THE THEORY THAT WOULD NOT DIE
How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines and Emerged Triumphant From Two Centuries of Controversy

Sharon Bertsch McGrayne introduces Bayes’s theorem in her new book with a remark by John Maynard Keynes: “When the facts change, I change my opinion. What do you do, sir?”

ayes’s theorem, named after the 18th-century Presbyterian minister Thomas Bayes, addresses this selfsame essential task: How should we modify our beliefs in the light of additional information? Do we cling to old assumptions long after they’ve become untenable, or abandon them too readily at the first whisper of doubt? Bayesian reasoning promises to bring our views gradually into line with reality and so has become an invaluable tool for scientists of all sorts and, indeed, for anyone who wants, putting it grandiloquently, to sync up with the universe. If you are not thinking like a Bayesian, perhaps you should be.

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: How confident am I in the truth of my initial belief? On the assumption that my original belief is true, how confident am I that the new evidence is accurate? And whether or not my original belief is true, how confident am I that the new evidence is accurate? One proto-Bayesian, David Hume, underlined the importance of considering evidentiary probability properly when he questioned the authority of religious hearsay: one shouldn’t trust the supposed evidence for a miracle, he argued, unless it would be even more miraculous if the report were untrue.
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