This is a homework exercise but I am stuck, and I believe there is something basic that is still confusing me. This is the problem:
There is a test for a new illness. The lab who developed did the following:
- To determine the false positive rate they tested 3000 known negative samples and got 15 positive results.
- To determine the false negative rate they tested 200 known positive samples and got 30 negative results.
- Then they tested 4000 people and got 60 positive results.
Determine the posterior distribution on the true incidence, marginalizing over false positive and false negative rates. Assume flat priors on all parameters.
My work
From point (1) I can easily get a posterior distribution on the false positive rate, $p_{fp}$:
$$ f(p_{fp}|\text{data}) = \frac{P(\text{data}|p_{fp})}{\int_0^\infty P(\text{data}|p_{fp})} $$ where $P(\text{data}|p_{fp})$ is just a binomial distribution with 15 successes, 3000 trials, and probability of success $p_{fp}$.
I can then do exactly the same and get a posterior distribution for the false negative rate. My problem is that I don't know how to move forward )use this information) to get the posterior on the true incidence.
Is this what I am supposed to compute? $$ P(N \text{ positive}|\text{60 positive out of 4000}) = \frac{P(\text{outcome}|\text{incidence})}{\sum_{n=0}^{4000}P(\text{outcome}|\text{incidence})} $$
If this is truly what I need to calculate, how do my posterior distributions on false-positive and false-negative used here?
I think I am a bit confused on how to proceed, but hopefully I am not too lost.