F. D. Flam’s essay “The Odds, Continually Updated” revisits the famous Monty Hall Problem and appears in this week’s Science Times. (The science section of the N.Y. Times, which appears every Tuesday, offers a fascinating glimpse into the wonderful world of contemporary science.) Here is an excerpt from Flam’s excellent essay (emphasis ours):
The essence of the frequentist technique is to apply probability to data. If you suspect your friend has a weighted coin, for example, and you observe that it came up heads nine times out of 10, a frequentist would calculate the probability of getting such a result with an unweighted coin. The answer (about 1 percent) is not a direct measure of the probability that the coin is weighted; it’s a measure of how improbable the nine-in-10 result is — a piece of information that can be useful in investigating your suspicion.
By contrast, Bayesian calculations go straight for the probability of the hypothesis, factoring in not just the data from the coin-toss experiment but any other relevant information — including whether you’ve previously seen your friend use a weighted coin.
Scientists who have learned Bayesian statistics often marvel that it propels them through a different kind of scientific reasoning than they’d experienced using classical [i.e. frequentist] methods. “Statistics sounds like this dry, technical subject, but it draws on deep philosophical debates about the nature of reality,” said the Princeton University astrophysicist Edwin Turner, who has witnessed a widespread conversion to Bayesian thinking in his field over the last 15 years.
Deborah Mayo reviews Flam’s essay here and retorts: “Yes, but … what many would like to know is how to cross check Bayesian methods—how do I test your beliefs?” We will give this important question some thought and respond in a future blog post (it’s way past our bedtime). In the meantime, whether you are a frequentist or a Bayesian (or something else altogether), why aren’t we teaching school children (or lawyers, for that matter) about the concepts of probability and error as well as the rudiments of probability theory?