October 21, 2013 Leave a comment
“You are suffering from a disease that, according to your manifest symptoms, is either A or B. For a variety of demographic reasons disease A happens to be nineteen times as common as B. The two diseases are equally fatal if untreated, but it is dangerous to combine the respectively appropriate treatments. Your physician orders a certain test which, through the operation of a fairly well understood causal process, always gives a unique diagnosis in such cases, and this diagnosis has been tried out on equal numbers of A- and B-patients and is known to be correct on 80% of those occasions. The tests report that you are suffering from disease B. Should you nevertheless opt for the treatment appropriate to A … ?”
My thoughts below …
If, following Good, we use
P(A|B:C) to denote the odds of A, conditional on B in the context C, Odds(A1/A2|B:C) to denote the odds P(A1|B:C)/P(A2|B:C), and LR(B|A1/A2:C) to denote the likelihood ratio, P(B|A1:C)/P(B|A2:C).
then we want
Odds(A/B | diagnosis of B : you), given
Odds(A/B : population) and
P(diagnosis of B | B : test), and similarly for A.
This looks like a job for Bayes’ rule! In Odds form this is
Odds(A1/A2|B:C) = LR(B|A1/A2:C).Odds(A1/A2:C).
If we ignore the dependence on context, this would yield
Odds(A/B | diagnosis of B ) = LR(diagnosis of B | A/B ).Odds(A/B).
But are we justified in ignoring the differences? For simplicity, suppose that the tests were conducted on a representative sample of the population, so that we have Odds(A/B | diagnosis of B : population), but still need Odds(A/B | diagnosis of B : you). According to Blackburn’s population indifference principle (PIP) you ‘should’ use the whole population statistics, but his reasons seem doubtful. Suppose that:
- You thought yourself in every way typical of the population as a whole.
- The prevalence of diseases among those you know was consistent with the whole population data.
Then PIP seems more reasonable. But if you are of a minority ethnicity – for example – with many relatives, neighbours and friends who share your distinguishing characteristic, then it might be more reasonable to use an informal estimate based on a more appropriate population, rather than a better quality estimate based on a less appropriate estimate. (This is a kind of converse to the availability heuristic.)
My notes on Cohen for a discussion of alternatives.
Other, similar, Puzzles.
My notes on probability.