Bayes rule with multiple tests

Question

Say you are given the following

\begin{align*} P(cancer) &= 0.008, & P(\neg cancer) &= 0.992\\ P(\oplus|cancer) &= 0.98, & P(\ominus|cancer) &= 0.02\\ P(\oplus|\neg cancer) &= 0.03, & P(\ominus|\neg cancer) &= 0.97 \end{align*}

and you test a patient to see if they have cancer or not and it returns positive. If the patient wants to be tested another time, and it once again returns positive, what happens to the probability that the patient has cancer after testing a second time?

Answer 1

According to Bayes' Theorem, \begin{align*} P(\text{cancer} \mid \oplus \& \oplus) &= \frac{P(\text{cancer} \cap \oplus \& \oplus)}{P(\oplus \& \oplus)}\\ &= \frac{P(\oplus \& \oplus \mid \text{cancer})P(\text{cancer})}{P(\oplus \& \oplus \mid \text{cancer})P(\text{cancer})+P(\oplus \& \oplus \mid \neg\text{cancer})P(\neg\text{cancer})} \end{align*} Now we have to make some assumptions about the two tests, and I think it is reasonable to assume their outcomes are independent. E.g., if someone has cancer, then the second test is positive with $98\%$ chance regardless of the first test. Then we can say: $$P(\oplus \& \oplus \mid \text{cancer})=P(\oplus \mid \text{cancer})P(\oplus \mid \text{cancer})$$ and similarly $$P(\oplus \& \oplus \mid \neg\text{cancer})=P(\oplus \mid \neg\text{cancer})P(\oplus \mid \neg\text{cancer}).$$ You should be able to finish from here.

Answer 2

There isn't simply enough information to answer this. Sure, you could think that the results of the two tests are conditionally independent given that the patient has a cancer or not. However, in most cases, they are not and usually whoever results positive to a test tend to result positive to other tests whether or not they have a cancer. This because different tests might react to a condition which is highly correlated with the desease.

Say for instance that the presence of a given hormone (or whatever) is highly indicative of a desease (however, someone could have the hormone and be totally healthy and someone else with the desease might not develop the hormone). Then, you have two tests that, with different tecniques, aim to detect the presence of the hormone (both tests might be wrong sometimes). If someone results positive to a test, the presence of the hormone becomes more probable, and consequently, the patient is more likely to have the desease. With the second positive test, the hormone still becomes more likely to be there. However, this won't enhance much the probability that the patient has the desease, since they might be someone of the healthy guys with the hormone.

So, either you state that the tests are conditionally independent (and so the other answer you received can be applied) or you can't respond to your question.