I have just started reading Daniel Kahneman’s book Thinking, Fast and Slow. In the introduction he gives this example (my précis).
Steve is neat and orderly and good with detail. Is he more likely to be a farmer or a librarian?
Since we “know” these characteristics belong to a librarian, most people would say “librarian.” Kahneman calls this type of reasoning, the resemblance heuristic. In Steve’s case he argues that since there are about twenty farmers to every one librarian, Steve is more likely to be a farmer. Surely more than one out of twenty farmers could be described as neat and orderly.
This is an example of Bayesian analysis (I am sure Kahneman will say so later in the book.) Most people ignore or never think to use the a priori probability of 20 to 1 farmers to librarians. If they had, much more evidence would be needed, say that Steve has memorized the Dewey Decimal System, to prove he was a librarian.
Here is another example of Bayesian reasoning. If I were to be diagnosed with a rare tropical disease, I would want plenty of evidence that I had the disease before undertaking a potential hazardous drug regime since I have never been in a tropical climate. My a priori probability of having the disease is very low. This seems just common sense.
When I teach elementary statistics from the traditional frequency point of view, I often feel that my students are just going through the motions and really don’t buy into it. Is that because we are all instinctively though imperfectly Bayesians at heart?