It certainly isn't. The probability of data given a hypothesis, P(D|H), is not the same as the probability of the hypothesis given the data, P(H|D). This is an elementary error regardless of one's preferred statistical approach. Bayesian statistical inference, which includes syllogistic reasoning as a special case, is particularly well-suited for avoiding this sort of mistake [2].
A similar pitfall is the infamous ``prosecutor's fallacy'', in which a probability that a DNA fingerprint match would occur in someone other than the true criminal -- P(match|innocent) -- is used incorrectly as the probability that a suspect is innocent -- P(innocent|match). In a city of ten million, a one in a million DNA fingerprint match will give ten other people the same fingerprint as the true criminal. In the absence of other evidence, the odds that the suspect is innocent are better than 90%, not one in a million.
Let H represent the class (hypothesis) of human-ness, A represent the class (hypothesis) of alien-ness, and J represent the observation (data) that a randomly chosen individual is Pope John Paul II. Bayes' theorem tells us how to infer the probability that the Pope is human, P(H|J):
P(H|J) = P(J|H) P(H) --------------------- P(J|H)P(H) + P(J|A)P(A)Thus, to infer the probability that the Pope is human, P(H|J), we have to have two more numbers in addition to the probability P(J|H) of drawing the Pope at random from the class of humans:
1) P(J|A), the probability of choosing the Pope at random from the class of aliens.
2) P(A), the a priori probability that a randomly chosen individual is an alien instead of a human. P(H) is just 1 - P(A), if only these two hypotheses are considered.
Presumably the probability P(J|A) of choosing the Pope from the class of aliens is infinitesimal. The prior probability of choosing an alien as opposed to a human, P(A), is also expected to be quite small, except perhaps near secret U.S. Air Force bases. As either P(J|A) or P(A) approach zero, the probability that the Pope is human approaches one.
It is a shame that Bayesian methods are not part of all introductory statistics classes. In this case, Bayesian methods quickly reassure us that the Pope is (probably) not an alien.
Sean R. Eddy Dept. of Genetics, Washington University School of Medicine, 660 S. Euclid Box 8232, St. Louis MO 63110, USA
David J.C. MacKay Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK
1. Nature 381,730 (1996).
2. Jeffreys, H. Theory of Probability (Oxford University Press,
Oxford UK, 1939).