I realize now that I misunderstood Mayo’s use of J in place of ~H. ~H says that breast cancer is absent, whereas J says that breast disease (inclusive of breast cancer) is absent. The idea seems to be that a non-cancerous breast disease is likely to trigger a false-positive result in a test for breast cancer, and that this possibility makes it the case that a positive test result does not pass H severely.
This point does not affect the upshot of my analysis. Howson can simply stipulate a hypothetical case in which there is no state corresponding to J in which a false positive test is likely. That is enough to show that the severity requirement is unsound in principle.
Moreover, Mayo grants that ~J (which says that breast disease is present) does pass a severe test despite having (we can assume) a low posterior probability. Thus, she allows that a hypothesis can meet the severity requirement despite having a low posterior, effectively granting premise 2 of Howson’s argument, and turns her attention to premise 4.
Here is Mayo’s reconstruction of Howson’s argument modified to reflect the fact that Mayo denies that H passes a severe test but allows that ~J does so:
- An abnormal result e is taken as failing to reject H (i.e., as “accepting H”); while rejecting J, that no breast disease exists
- ~J passes a severe test and thus ~J is indicated according to (*). (Modified)
- But the disease is so rare in the population (from which the patient was randomly sampled) that the posterior probability of ~J given e is still very low (and that of ~H is still very high). (Modified)
- Therefore, “intuitively,” ~J is not indicated but rather ~H is. (Modified)
- Therefore, (*) is unsound.
Mayo’s argument against premise 4 is interesting, but an orthodox Bayesian has an easy response. She points out that both error-statistical and Bayesian tests involve probabilistic calculations that are themselves deductive. The error-statistical framework only becomes ampliative with the introduction of the severity requirement (*), which goes beyond those calculations to make an assertion about which claims are well supported by tests. She demands that (*) be compared not against the deductive probabilistic calculations that Bayesians perform, but against a truly ampliative Bayesian rule. What she is demanding, in effect, is a rule of detachment, which tells a Bayesian when to infer from a statement of the form “the probability of H is p” to the statement “H.”
A Bayesian has at least two possible responses to this maneuver. First, it is not clear that Bayesian updating is a deductive method of inference. It uses a rule—Bayes’ theorem—that follows from the axioms of probability, but those axioms are not dictated by classical logic, nor is the normative claim that the right way to update one’s degree of belief in H when one has an experience the only direct epistemic import of which is to change one’s degree of belief in E to 1 is by conditioning on E, for all propositions H and E. Second, Mayo has not given any reason why Bayesians should adopt a rule of detachment, rather than being strict probabilists. The demand that orange-selling Bayesians provide an apple to compare with her apple would be unfair if part of the Bayesian position were that oranges can do everything apples can do at least as well apples do it. (A Bayesian can still approximate high-probability beliefs as full beliefs as a useful heuristic when doing so is not likely to lead to trouble.)
Having demanded (unfairly) that Bayesians adopt a rule of detachment, Mayo ascribes to Howson the following implicit rule:
· There is a good indication or strong evidence for the correctness of hypothesis H just to the extent that it has a high posterior probability.
She then turns Howson’s example against him, ascribing to him this rule of detachment. She notes that for a woman in her forties, the posterior probability of breast cancer given an abnormal mammogram is about 2.5%, which makes it very close to Howson’s hypothetical example. Under an error-statistics approach, the hypothesis that such a woman does not have breast cancer does not pass a severe test with a positive result; nor does the hypothesis that such a woman does have breast cancer. To provide strong evidence one way or the other, follow-up tests are needed. Under a Bayesian approach with a rule of detachment, the fact that the posterior probability that the woman has breast cancer is small provides a good indication that breast cancer is absent, “so the follow-up that discovered these cancers would not have been warranted.”
This argument is grossly unfair to the Bayesian position. For an orthodox Bayesian, whether or not follow-up tests are warranted (and whether or not the initial test was warranted) for a given individual depends on that individuals expected utilities. Rounding down probability 2.5% to 0% in an expected utility calculation is likely to lead to errors when the utility of the unlikely event is extremely high or extremely low, as in this case. This fact speaks not against Bayesianism, but against simple rules of detachment.
In summary, Mayo has shown that error statistics is sometimes more sensible than Bayesianism with a simple-minded rule of detachment. But a sensible Bayesian would not use such a rule of detachment, so this conclusion has no force against Bayesianism.