Reading Ron Chernow’s magisterial new biography of Ulysses Grant, I came across this very correct statistical inverse reasoning from the celebrated journalist Horace Greeley (whose role in the high school history curriculum has been reduced to the phrase, “Go West, young man” — that he denied having invented):
All Democrats are not horse thieves, but all horse thieves are Democrats.
This seems like an ironic bon mot, but after he became the Democratic candidate for president against Grant in 1872 he tried to use a milder version unironically as a defence of his new party colleagues:
I never said all Democrats were saloon keepers. What I said was all saloon keepers are Democrats.
Presumably he meant to add that if we knew the base rate of saloonkeeping (or horse thievery) in the population at large, we could calculate from the Democratic vote share the exact fraction of Democrats (and of Republicans) who are saloonkeepers (or horse thieves).
I was reading (finally, after seven years in Oxford) Thomas Hardy’s Jude the Obscure, and found the following quote from John Henry Newman:
My argument was … that absolute certitude as to the truths of natural theology was the result of an assemblage of concurring and converging probabilities … that probabilities which did not reach to logical certainty might create a mental certitude.
I was talking recently to a friend who said he saw the story of the Trojan horse as an object lesson in the failure of governance. “Wasn’t there anyone who could say, wait a minute, maybe it’s just not a good idea to bring that horse in here, even if the Greeks seem to have all left?”
I said it was a fable about the inaccessibility of Bayesian reasoning. Laocoön warned them that the prior probability for a net benefit from a Greek gift was low (timeō Danaōs et dōna ferentēs). But the Trojans placed more credence in new information, particularly private information that they hold exclusively, particularly when they seem to have won the information at great effort, by their own ingenuity, by torturing the captured Sinon. (This lesson was learned by the British spies in WWII who conceived Operation Mincemeat.) Laocoön was punished for insisting on his strong prior, being crushed to death by the clever serpents sent by the Goddess of Worldly Wisdom. And the Trojans celebrated their ingenious victory, until they were overrun by reality, in the form of well-muscled Achaean warriors who were not impressed by their highly significant rejection of the likelihood of a subterfuge.