Today's New York Times has an article on "a once obscure field known as Bayesian statistics."1/ It is an informative piece by Faye Flam, a science journalist with an uncommonly good grasp of science. But a quantum of confusion infects the effort to contrast "Bayesian statistics" with "the more traditional or 'classical' approach, known as frequentist statistics."
The article presents the solution to famous Monty Hall problem (known to "classical" probabilists as the three-curtains problem long before its appearance in the TV game show) as especially amenable to "Bayesian statistics." But frequentist thinking works quite well here. In the long run, the strategy of switching beats the strategy of not switching. This is easily proved with classical, objective probabilities.
Indeed, it is not clear that the Monty Hall problem is even a problem in statistical inference.2/ There are no statistical (sample) data to consider and no sense in which the use of Bayes' rule to solve the probability problem "counter[s] pure objectivity." How, then, do "[t]he two methods approach the same problem from different angles"?
Of course, the Monty Hall problem is nice for illustrating the power of Bayes' rule in working with conditional probabilities. I have used it in this way in my courses, and that may have been the reason it appears in the article. But it does not illustrate the philosophical divide between frequentists and Bayesians.
To this extent, it is disappointing that the Times (but probably not the author) chose to start the online version of the article with a large photograph of Monty Hall captioned "
1.Faye D. Flam, The Odds, Continually Updated, N.Y. Times, Sept. 30, 2014, at D1.
2. On the distinction between a "problem of statistical inference or, more simply, a statistics problem," and a probability problem, see, for example, Morris H. DeGroot, Probability and Statistics 257 (1975).