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DanielLC comments on From First Principles - Less Wrong Discussion
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Insuffiient information. For example, if the base rate is 80% and the positive rate is 80%, it could be anything from a perfect test to a random result independent of them having cancer. You need another equation, like knowing the false positive rate, or knowing that the false positive and false negative rates are the same.
Which isn't to say that this would be a bad test question, of course - just that the student has to realize that they're expected to explain why they have insufficient information to answer.
Well you can at least give the liklihood ratio of a positive test, which was what I was getting at. You're right tho, to give all the test parameters, you'd need the negative rate as well.
Maybe conservation of evidence can help us fill in the blanks somewhere? This seems like a fun thing to think about.
How? You need to know how likely it is to be positive given that you have cancer. If it's a perfect test, the likelihood ratio is infinity to one. If it's a random test, it's one to one. Since it could be either of those, or anything in between, you can't figure that out.
Am I misinterpreting what you mean by positive rate, likelihood ratio, or both?
prior odds are 1:99 against, posterior odds are 1:9.9, therefore LR of positive is 10. I may have miscommunicated. I meant posterior when I said "positive rate" ("cancer rate given positive" was my interpretation) I can see how it is better parsed as "rate that you test positive", which is something else. Sorry for the confusion.