# Probabilities in case-controlled studies

I have a nested-case control study that I have been using for analysis. At the end of my work I have deduced a set of variables that I use later to to classify new cases. One example of a simple classifier I am using is a naive Bayes, which will output simply a probability.

So here is my question:

Could I make my probabilities reflect the real world? In my specific example, the condition that I am testing for has a prevalence of 33% in my study, but a it has a population prevalence of only 10%. Bayes factors have been suggested to me as a way to achieve this, however I am little unsure how to set up the problem.

As an example I have seen a Bayes factor as a logit between the true vs. study prevalence of the outcome. The classifier however was a logistic regression, and in that case the Bayes factor was just added to the linear predictors. I think the example there was very specific, and perhaps an inappropriate method for probabilities of a naive Bayes. Instead what I did was add the logit Bayes factor to the logged probabilities, but I am also not convinced this is right either. I also think a simpler solution would be to use Bayes theorem directly, but there I am not sure how to represented my study vs.population prevalences. The method below isn't quite right, but gets at what I want:

        p_final = classier_posterior*(population_prev)/(study_prev)


I should contextualize that I use the probabilities to establish a threshold for classification down stream.

$P(C|F_1,\ldots,F_n) \propto P(C) \prod_{i=1}^n P(F_i|C)$
The $P(F_i|C)$ terms are estimated from the data, but instead of estimating $P(C)$ from the data (study prevalence), you use a different measure (population prevalence). This is identical to your proposal above.