In a post last year, I went through some inference problems concerning a hypothetical medical test. For example, using the known rate of occurrence of some disease, and the known characteristics of a diagnostic test (false-positive and false-negative rates), we were able to obtain the probability that a subject has the disease, based on the test result.
In this post, I'll demonstrate some hierarchical modeling, in a similar context of medical diagnosis. Suppose we know the characteristics of the diagnostic test, but not the frequency of occurrence of the disease, can we figure this out from a set of test results?
A medical screening test has a false-positive rate of 0.15 and a false-negative rate of 0.1. One thousand randomly sampled subjects were tested, resulting in 213 positive test results. What is the posterior distribution over the background prevalence of the disease in this population?