Another stupid question about Bayes Theorem.

Let's say I go to the doctor and take some sort of screening test for cancer. Only 1% of the population has this cancer, and this test has an 80% success rate and 10% false positive rate. The test says I have the cancer, so this means that I'm now at approx. 7% probability that I have the cancer. If I go to a different doctor's office the next day and take the same test, am I updating on the original 1% or am I updating on the new 7%?

You update on the 7%, but the conditional probabilities of getting a positive result if you have and not have cancer respectively have also changed so that another positive test gives you less information than what it took to bring you from 1% to 7 %. See what ygert said.

4ygert7yIt depends on the test, that is to say that the information given is not enough to determine it. If different instances of the test are independent, then you update on the 7%. If they are partially or completely dependant on one another, it is a lot more complicated. (An analogy: If you roll a die, there is a 1/2 chance of getting an even number, and a 1/2 chance if getting an odd number. If you roll the die twice, there is a 1/2 * 1/2 =1/4 chance of the first roll being odd, and the second roll being even, because the rolls are independent. Between non-independent events, you cannot just multiply out the probabilities like that. The chance of getting both an odd number and an even number on one die roll is 0, despite the fact that on a surface level it seems the same as the previous example: Two tests, each with a 1/2 chance of passing.) So again, it depends on the type of test. If, say, the test is that people with a certain gene have more of a chance for the cancer (the test can always find the gene, and tells you the gene with 100% accuracy, but while having the gene increases your risk, it isn't absolute, giving the numbers you gave), then it is obvious that the new test will give you no new information. You know the test will again say that you have the gene, and you stay at the 7% confidence level. On the other hand, if the test is, say, some sort of scan, that will look for precancerous tissue, and in each unit of time, it has a certain (fixed) chance of either finding precancerous tissue if it exists or of falsely finding precancerous tissue, then multiple instances of the test will be independent, and so you can "add the information together", first updating on one test, then on the other, so you will do the update from the new test from your 7% confidence level. (Actually, in real life, it will be far more complicated than that, with different test being only partially independent. It is unrealistic to think the first test will correctly identify the ge
2asr7yIt's not a stupid question. But I believe the answer depends on the details of the test. For some tests, the probabilities on successive tests will be correlated, and for others, they won't. My impression is that part of what people learn when they become medical specialists is when it makes sense to repeat the same test, and when it makes sense to look for a different test with uncorrelated errors.

More "Stupid" Questions

by NancyLebovitz 1 min read31st Jul 2013498 comments

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This is a thread where people can ask questions that they would ordinarily feel embarrassed for not knowing the answer to. The previous "stupid" questions thread went to over 800 comments in two and a half weeks, so I think it's time for a new one.