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DanielLC comments on From First Principles - Less Wrong
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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.