Saturday, January 29, 2011

Mayo's Reconstruction of Howson's Argument

Mayo’s response to Howson* has many threads.  I will focus on the passages in which she responds most directly to Howson’s challenge, beginning with Mayo’s reconstruction of Howson’s argument.  Mayo makes Howson's example more concrete by specifying that the disease being tested for is breast cancer.
  • e: An “abnormal result,” in this case a positive test result (i.e., one that provides at least incremental evidence of breast cancer).
  • H: The hypothesis that the patient has breast cancer.
  • J: The hypothesis that breast disease is absent
  • (*): The claim that e is a good indication of H to the extent that H has passed a severe test with e.  (Mayo’s “severity requirement”)

Why does Mayo use J for the hypothesis that the disease is absent instead of ~H?  Well, she points out that “not-H” is a disjunction of many different hypotheses and claims that in order to calculate error probabilities, we need to consider specific alternatives that fall under the heading “not-H.”  It is certainly true that in realistic cases involving diseases ~H could be expressed as a disjunction of more specific alternatives.  In general, however, those alternatives could themselves be expressed as disjunctions of even more specific alternatives ad infinitum.  Thus, simply the fact that ~H could be broken down further does not entail that ~H cannot supply error probabilities.  In a hypothetical example such as Howson’s, there is no reason why one cannot simply stipulate that error probabilities for ~H are available.  Moreover, the hypothesis J, that the disease is absent, is no more specific than the negation of H.  It just is the negation of H.  As a result, Mayo’s use of J here seems to me a needless complication.  Nevertheless, I will follow her notation to lessen the risk of misrepresenting her argument.

UPDATE: I misunderstood Mayo here.  J indicates the absence of breast disease of any kind, not just breast cancer, so it is not equivalent to ~H.  See subsequent posts.

With this notation in place, Mayo reconstructs Howson’s argument as follows: 
  1. An abnormal result e is taken as failing to reject H (i.e., as “accepting H”); while rejecting J, that no breast cancer exists.
  2. H passes a severe test and thus H is indicated according to (*).
  3. But the disease is so rare in the population (from which the patient was randomly sampled) that the posterior probability of H given e is still very low (and that of J is still very high).
  4. Therefore, “intuitively,” H is not indicated but rather J is.
  5. Therefore (*) is unsound.

This reconstruction illustrates a common practice in philosophy that I find annoying of presenting in a non-deductive format an argument that could easily and without distortion be presented in a deductive format.  I do not believe that all arguments should be understood deductively (Musgrave-style deductivism), or that all philosophical projects should have at their core a list of premises followed by a conclusion that follows from those premises either deductively or inductively (although I tend to be drawn toward such projects personally).  I am simply saying that when one is either advancing or criticizing an argument that can be put into a deductive format without distortion, it is a salutary practice to present the argument in such a format.  Doing so helps to clarify exactly what conditions must be met for the conclusion to go through.  But I digress.  I think Mayo's reconstruction is adequate for what follows, but I'll keep an eye on how her response fares relative to my own deductive reconstruction.

Mayo rejects premise 2, which she says is a misapplication of her severity requirement (*).  This premise corresponds well to premise 1 from my deductive reconstruction of Howson's argument.  Mayo also rejects premise 4 of her reconstruction, which “assumes the intuitions of Howson’s Bayesian rule.”  This premise corresponds most closely to premise 6 in my reconstruction, although elements of my premises 4 and 5 may be implicated as well.  

Mayo begs the question somewhat in her premise 4 by implying that Bayesianism rests on a bare appeal to intuition—Bayesians could point to a variety of sophisticated arguments that one ought to have degrees of belief that accord with the probability calculus and update those degrees of belief according to Bayes’ theorem.  Whether those arguments succeed or not is a separate question, of course, but they do deserve to be taken seriously.

In my next post, I plan to consider Mayo’s grounds for rejecting premises 2 and 4 of her reconstruction.

*The article I am considering is Deborah  Mayo’s 1997 “Error Statistics and Learning from Error: Making a Virtue of Necessity.”  It appeared in Philosophy of Science  Vol. 64, Supplement.  Proceedings of the 1996 Biennial Meeting of the Philosophy of Science Association.  Part II: Symposia Papers (Dec., 1997), pp. S195-212.  It is a response to Colin Howson’s “Error Statistics in Error,” pp. S185-S194 in the same issue.

No comments:

Post a Comment