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Alicorn comments on Mammography problem from 'Intro to Bayes' in my own words/picture - Less Wrong
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As far as the take-home practical message goes, on my reading it was never about how well doctors could "diagnose cancer" per se based on mammogram results--rather, the reason we ask about P(cancer | positive) is because it ought to inform our decision about whether a biopsy is really warranted. If a healthy young woman from a population with an exceedingly low base rate for breast cancer has a positive mammogram, the prior probability of her having cancer may still be low enough that there might actually be negative expected value in following up with a biopsy; after all, let's not forgot that a biopsy is not a trivial procedure and things do sometimes go wrong.
So I think this actually does have some implication for real-world clinical care: we ought to question whether it is wise to automatically follow up all positive mammograms with biopsies. Maybe it is, and maybe it isn't, but I don't think we should take the question for granted as appears to be the case.
If a biopsy is the next step in diagnosing breast cancer after a positive mammogram, then we shouldn't perform mammograms on anyone it still wouldn't be worth biopsying should their mammogram turn up positive.
Yes, that's exactly right.
And although I'm having a hard time finding a news article to verify this, someone informed me that the official breast cancer screening recommendations in the US (or was it a particular state, perhaps California?) were recently modified such that it is now not recommended that women younger than 40 (50?) receive regular screening. The young woman who informed me of this change in policy was quite upset about it. It didn't make any sense to her. I tried to explain to her how it actually made good sense when you think about it in terms of base rates and expected values, but of course, it was no use.
But to return to the issue clinical implications, yes: if a woman belongs to a population where the result of a mammogram would not change our decision about whether a biopsy is necessary, then probably she shouldn't have the mammogram. I suspect that this line of reasoning would sound quite foreign to most practicing doctors.