Suppose that you have $N$ people passing an exam having 2 type of questions (type A and type B) which all have correct/uncorrect outcomes. After seeing the results, I want to modelize the probability of success for one person for every type of questions (A and B) after he has answer a few of this questions already.
I modelize the questions results by two bernoulli laws $X_1=B(p_1)$ and $X_2=B(p_2)$. Each person has his own $p_1, p_2$ but we think that they’re linked. I’m looking for a way to use this dependence …
So far, I have chosen an a priori law on $p_1, p_2$ : the beta$(1,1)$ function for both. We know that if a person answers with $S$ success and $E$ errors for the first type of questions, the a posteriori law will be beta$(1+S,1+E)$. But we don’t know anything about how it changes $p_2$. My guess is that $p_2$ could be something like beta$(1+kS,1+kE)$ with $k$ a coefficient representing how much the first type of question and the second type of question are correlated. This intuition comes from the facts that if the type of questions are independent then the a posteriori law of $p_2$ is still beta$(1,1)$ and if they are perfectly correlated, the a posteriori law of $p_2$ is beta$(1+S,1+E)$.
As someone told me in a previous post, the IRT theory could help but IRT seems to assume that each person has a fixed ability $\theta$ while for my problem, there is 2 abilities and I want to modelize their interdependence.
What about in between ? Any references/books/etc would be greatly appreciated.