Talk:Mix math and medicine and create confusion
It seems to me that this would be a good way to introduce the difference between the Bayesian and frequentist point of view.
From the frequentist point of view, one expects 60% of a large group of people to respond positively. However, the statement doesn't apply to any particular individual, since from this point of view a given patient will either respond positively or not, but one cannot talk about the probability of how an individual patient will respond.
From the Bayesian point of view, of course, it does make sense to talk about the probability that an individual will respond positively. This can be expressed in a number of ways, e.g., as a bet at certain odds that the patient will respond (indicating ones confidence that this will happen) or simply as the degree of belief that one holds that this will happen.
The patient really wants to know how to respond to the information given by the physician in her particular case. Thus, the patient really would like to interpret probability in a Bayesian sense, since she is the one whose health is on the line, and the behavior of a large number of people is not really relevant.
One might expand the discussion to talk about how the probability is conditional on other knowledge that the physician might have, e.g., whether the patient had relatives with the same condition, etc.
It does appear to be the case that in general, physicians are not very adept at making correct inferences in such cases. See Gerd Gigerenzer, Calculated Risks (New York, 2002)
The other aspect, mentioned by Friedman, is that the problem as stated isn't just an inference problem, but it is a decision problem. This means that the potential consequences of taking any particular action (including "doing nothing") must be taken into account. For example, a particular drug may have side effects, and if they are potentially severe, the decision to take the drug has to take that into account as well as the probability that the patient will respond positively.
For some years now I have been teaching a course on Bayesian inference and decision theory to incoming honors freshmen. We discuss these ideas in various contexts (business, medicine, law, science, etc.), keeping the ideas simple (finite state space) but challenging. Many of my students are premed, and I have begun to get feedback that they are finding these ideas useful in medical school.
Bill Jefferys 13:03, 30 May 2005 (EDT)