I reckon that utilizing Bayesian method often produces counterintuitive results, but the example below seems counterintuitive in an extent too extreme for me.
Suppose there is a specific disease and a company is trying to make a kit which can detect it, and the probability of such case is as below. ( "+" indicates that the kit has showed positive.)
|got Disease||did not get Disease|
Then, the equations go
P(+|D)= 0.9 [which indicates that the performance of the kit is decent in the perspective of the company];
yet, P(D|+)= 0.083 [which indicates that an individual who got the positive result do not have any reasonable evidence to believe she actually has a disease.]
I understand that this dilemma appeared because of the huge asymmetry between the probabilities, and I can grasp it mathematically by using diagrams. In addition, one might say that the probability of getting disease is updated and increased from .009 to .083 due to the happening of event (+).
However, considering that social science needs to interpret the numeric statistical results into a logical and intuitive explanations, the latter result to me is utterly unexplainable.
While the former result implies to the kit manufacturer that they have invented reliable kit, the latter - which is what we as individuals face - may intuitively tell people - say, those who are diagnosed positive with COVID test - that the possibility that they actually got infected is under 10%. On the basis of this explanation, perhaps there is no need for one to think that they should be hospitalized or get medically treated just because they got + sign on their kit.
Can this gap between mathematical result and logical(i.e. intuitive) explaination be harmonized?